Convergence Analysis of Distributed Set-Valued Information Systems

This paper focuses on the convergence of information in distributed systems of agents communicating over a network. The information on which the convergence is sought is not represented by real numbers, as often in the literature, rather by sets. The dynamics of the evolution of information across the network is accordingly described by set-valued iterative maps. While the study of convergence of set-valued iterative maps is highly complex in general, this paper focuses on Boolean maps, which are comprised of arbitrary combinations of unions, intersections, and complements of sets. For these important class of systems, we provide tools to study both global and local convergence. A distributed geographic information system, leading to successful information reconstruction from partial and corrupted data, is used to illustrate the applications of the proposed methods.

[1]  Sonia Martínez,et al.  Coverage control for mobile sensing networks , 2002, IEEE Transactions on Robotics and Automation.

[2]  Steven Givant,et al.  Introduction to Boolean Algebras , 2008 .

[3]  Keith Ansel Marzullo,et al.  Maintaining the time in a distributed system: an example of a loosely-coupled distributed service (synchronization, fault-tolerance, debugging) , 1984 .

[4]  S Sundaram,et al.  Control of quantized multi-agent systems with linear nearest neighbor rules: A finite field approach , 2010, Proceedings of the 2010 American Control Conference.

[5]  David L. Mills,et al.  Internet time synchronization: the network time protocol , 1991, IEEE Trans. Commun..

[6]  Shreyas Sundaram,et al.  Information dissemination in networks via linear iterative strategies over finite fields , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[7]  M. Stone,et al.  The Theory of Representation for Boolean Algebras , 1936 .

[8]  F. Robert.  Iterations sur des ensembles finis et automates cellulaires contractants , 1980 .

[9]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[10]  Stan Gudder.  BOOLEAN VECTOR SPACES , 2008 .

[11]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[12]  Antonio Bicchi,et al.  Distributed Consensus on Boolean Information , 2009 .

[13]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[14]  Antonio Bicchi,et al.  On the robust synthesis of logical consensus algorithms for distributed intrusion detection , 2013, Autom..

[15]  Ruggero Carli,et al.  Average consensus on networks with quantized communication , 2009 .

[16]  Andrea Garulli,et al.  Simultaneous localization and map building for a team of cooperating robots: a set membership approach , 2003, IEEE Trans. Robotics Autom..

[17]  Antonio Bicchi,et al.  Consensus Computation in Unreliable Networks: A System Theoretic Approach , 2010, IEEE Transactions on Automatic Control.

[18]  J. R. Buzeman.  Introduction To Boolean Algebras , 1961 .

[19]  F. Robert.  Dérivée Discrète et Comportement Local d'Une Itération Discrbte , 1983 .

[20]  M. Nosarzewska.  Évaluation de la différence entre l'aire d'une région plane convexe et le nombre des points aux coordonnées entières couverts par elle , 1948 .

[21]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[22]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[23]  Keith Marzullo,et al.  Maintaining the time in a distributed system , 1983, PODC '83.

[24]  Antonio Bicchi,et al.  Towards a Society of Robots , 2010, IEEE Robotics & Automation Magazine.

[25]  Leslie Lamport,et al.  The Byzantine Generals Problem , 1982, TOPL.