estimation in networks from relative...

Chenda Liao and Prabir Barooh

Time-Synchronization in Mobile Sensor

Networks from Difference Measurements

Distributed Control System Lab

Dept. of Mechanical and Aerospace Eng.

University of Florida, Gainesville, FL

49th IEEE Conference on Decision and Control

Dec, 15th, 2010

Atlanta, Georgia, USA

Sensor Networks

• Environment/Structure Monitoring

• Event/Fault Detection

• Home/office Automation

• Healthcare

• Industrial Automation

• Military Application

Limited power

=global/reference time =local time

=skew =offset

Time Synchronization in Sensor Network

Time synchronization problem is equivalent to determining and ,

Motivation:

Meaning of Sync. :

Global time

u

ref

v

Local time:

Literature review

Static sensor network

• Elson et al., Fine-grained network time synchronization using reference broadcasts (RBS), 2002

• Ganeriwal et al., Timing-Sync Protocol for Sensor Network (TPSN), 2003

• Barooah et al., Distributed optimal estimation from relative measurements for localization and time synchronization, 2006

• Suyong Yoon et al., Tiny-Sync: Tight Time Synchronization for Wireless Sensor Networks, 2007

Mobile sensor network

• Miklós et al., Flooding Time Synchronization Protocol (FTSP), 2004

• Su et.al., Time-Diffusion Synchronization Protocol for Wireless Sensor Networks (TDP), 2005

Noisy measurement of the

relative skews and offsets

for pairs of nodes

Time Sync on Mobile Sensor Network

Goal:

To estimate the skews and offsets of clocks of all the nodes with respect to an reference clock in mobile sensor network.

Algorithm:

• Model the time variation of the network (graph) as a Markov chain.

• Prove the mean square convergence of the estimation error (Markov jump linear system).

• Corroborate the predictions using Monte Carlo simulations.

Pair-wise synchronization: Network-wise synchronization:

Each node estimates its offset/skew from

noisy measurements by communicating

only with its neighbors iteratively.

Measurement Algorithm

• Directly measure and ? No!

• But node u can measure and .

• Method: pairs of nodes exchange time-stamped packet with two rounds communication.

u

v

Measurement

Algorithm

Noisy measurementTime-stamp Gaussian zero mean

Details in [Technical Report] Liao et al. Time Synchronization in Mobile Sensor Networks from Relative Measurements

Relative Measurement

ErrorsNode variablesNoisy

measurements

Same formulation for relative skew and offset measurement

Where

Relative

measurement

Reference

variable =0

Task:

Estimate all of the unknown nodes

variables from the measurements

and the reference variables.

Distributed Algorithm for Estimation

Algorithm:

Remark:

Where is the estimate of node variable at time k

is the neighbor set of node u at time k

is the number of neighbors in

• , if for all u by using flagged initialization.

• If node u has no neighbor, then .

Example:

4

3

6

Similar to an algorithm

in time sync of static

sensor network.

(e.g., Barooah et.al., 06)

• If evolution of satisfies Markovian property

• can be modeled as the realization of a Markov chain.

• Example

Performance Analysis

• Consider the graphs with 25 nodes, the number of different graphs is

State space and

An edge exists if

Euclidean

distance between

pair of nodes is

less than certain

value.

Position

Projection

function

Random

vector

• and are governed by Markov chain,

•

• is W.S.S, and

• and

Convergence Analysis

Theorem:

mean square

convergent

• is entry-wise positive

• At least one element of is a

connected graph

Conjecture:

• Markov chain is ergodic

• The union of all the graphs

in is a connected graph

Discrete-time Markov Jump Linear system modeling

calculable

mean square

convergent

Proof in [Technical Report] Liao et al. Time Synchronization in Mobile Sensor Networks from Relative Measurements

Simulation (4 nodes)

4 snapshots Mean of estimates of variables of node 3

Variance of estimates of variables of node 3

Number of edges of node 3 along timeSample trajectory of estimates of node 3

Simulation (25 nodes)

4 snapshots Mean of estimates of variables of node 25

Variance of estimates of variables of node 25

Number of edges of node 25 along timeSample trajectory of estimates of node 25

Simulation for Conjecture

All three possible graphs

Mean of estimates of variables of node 3 Variance of estimates of variables of node 3

Number of edges of node 3 along timeSample trajectory of estimates of node 3

Summary and Future Work

• A distributed time-synchronization protocol for mobile sensor networks.

• Mean square convergent under certain conditions.

Summary:

• Weaker sufficient conditions for mean square convergence

• Convergence rate in terms of the Markov chain's properties.

Future work:

Thank you!

This work has been supported by the National Science Foundation by Grants CNS-0931885 and ECCS-0955023.

Back up

Start from here

Algorithm for Measuring Offset

• Suppose , then

• Goal: Measure relative clock offset

• Method: Exchange time-stamped packet with a round trip.

•Let denote the measurement node u can obtain and

denote measurement error with mean 0 and variance

Where and are Gaussian i.i.d. random delay with mean and variance

•Finally, we obtain noisy measurement of relative clock offset (4)

Algorithm for Measuring Skew and Offset

• Now, consider

• Goal: Estimate both relative clock offset and skew

• Method: Exchange time-stamped packet with two round trips.

Estimating Relative Skew

• From (6), (7), (8) and (9), we obtain

(10)

Finally, after taking log on both side, (10) can be simplified as

(11)

Here, and

where

We make reasonably large by extending and shortening .

Then, implement Taylor series, we get . Therefore, it is not hard to

obtain and .

Estimating Relative Offset

• Reusing first four time stamp equations, we obtain

(12)

Again, let and .

Express (12) as

Distributed Algorithm

①

②

Sphere Graph

Position evolution:

The projection function:

Formula of Steady State Value

Let be the state space of real matrices.

Let be the set of all N-sequences of real matrices

The operator and is defined as follows:

Let

then , where and

For , then and

Then, let

Define and