Almost sure convergence of a randomized algorithm for relative localization in sensor networks

This paper regards the relative localization problem in sensor networks. We study a randomized algorithm, which is based on input-driven consensus dynamics and involves pairwise “gossip” communications and updates. Due to the randomness of the updates, the state of this algorithm ergodically oscillates around a limit value. Exploiting the ergodicity of the dynamics, we show that the time-average of the state almost surely converges to the least-squares solution of the localization problem. Remarkably, the computation of the time-average does not require the sensors to share any common clock. Hence, the proposed algorithm is fully distributed and asynchronous.

[1]  Angelia Nedic,et al.  Asynchronous Broadcast-Based Convex Optimization Over a Network , 2011, IEEE Transactions on Automatic Control.

[2]  R. Tempo,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems , 2004 .

[3]  P.R. Kumar,et al.  Distributed Clock Synchronization over Wireless Networks: Algorithms and Analysis , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[4]  Mario Sznaier,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems with Applications, Second Edition, Roberto Tempo, Giuseppe Calafiore, Fabrizio Dabbene (Eds.). Springer-Verlag, London (2013), 357, ISBN: 978-1-4471-4609-4 , 2014, Autom..

[5]  Roberto Tempo,et al.  Distributed Randomized Algorithms for the PageRank Computation , 2010, IEEE Transactions on Automatic Control.

[6]  Ruggero Carli,et al.  A PI controller based on asymmetric gossip communications for clocks synchronization in wireless sensors networks , 2011, IEEE Conference on Decision and Control and European Control Conference.

[7]  V. Borkar.  Probability Theory: An Advanced Course , 1995 .

[8]  Asuman E. Ozdaglar,et al.  Opinion Fluctuations and Disagreement in Social Networks , 2010, Math. Oper. Res..

[9]  Jia-An Yan,et al.  Semimartingale Theory and Stochastic Calculus , 1992 .

[10]  David C. Moore,et al.  Robust distributed network localization with noisy range measurements , 2004, SenSys '04.

[11]  Giacomo Como,et al.  Scaling limits for continuous opinion dynamics systems , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[12]  J. Hespanha,et al.  Estimation on graphs from relative measurements , 2007, IEEE Control Systems.

[13]  Nikolaos M. Freris,et al.  Fast distributed smoothing of relative measurements , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[14]  Damiano Varagnolo,et al.  Consensus‐based distributed sensor calibration and least‐square parameter identification in WSNs , 2010 .

[15]  Chiara Ravazzi,et al.  A distributed randomized algorithm for relative localization in sensor networks , 2013, 2013 European Control Conference (ECC).

[16]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[17]  Persi Diaconis,et al.  Iterated Random Functions , 1999, SIAM Rev..

[18]  Fabio Fagnani,et al.  Transient and limit performance of distributed relative localization , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[19]  João Pedro Hespanha,et al.  Error Scaling Laws for Linear Optimal Estimation From Relative Measurements , 2009, IEEE Transactions on Information Theory.

[20]  Er-Wei Bai,et al.  A Web Aggregation Approach for Distributed Randomized PageRank Algorithms , 2012, IEEE Transactions on Automatic Control.

[21]  João Pedro Hespanha,et al.  Estimation From Relative Measurements: Electrical Analogy and Large Graphs , 2008, IEEE Transactions on Signal Processing.

[22]  Er-Wei Bai,et al.  PageRank computation via a distributed randomized approach with lossy communication , 2012, Syst. Control. Lett..