estimation in networks from relative...
TRANSCRIPT
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