Distributed synchronization under uncertainty: A fuzzy approach

In this paper the synchronization of a network of identical systems with fuzzy initial conditions is introduced, as a convenient framework to obtain a shared estimation of the state of a system based on partial and distributed observations, in the case where such a state is affected by ambiguity and/or vagueness. After discussing the synchronization of crisp systems, providing a criteria to find the information sharing law that lets the network converge to a shared trajectory, Discrete-Time Fuzzy Systems (DFSs) are introduced as an extension of scalar fuzzy difference equations. Besides providing a stability condition for a general DFS, in the linear case it is proven that, under a non-negativity assumption for the coefficients of the system, the fuzzyfication of the initial conditions does not compromise the stability of the crisp system. A framework for the synchronization of arrays of linear DFS is then introduced, proving that the crisp synchronization is a particular case of the proposed approach. Finally, a case study in the field of Critical Infrastructure Protection is provided.

[1]  Stefano Panzieri,et al.  Online Distributed Interdependency Estimation for critical infrastructures , 2011, IEEE Conference on Decision and Control and European Control Conference.

[2]  Roberto Setola,et al.  Critical infrastructure dependency assessment using the input-output inoperability model , 2009, Int. J. Crit. Infrastructure Prot..

[3]  D. W. Pearson.  A property of linear fuzzy differential equations , 1997 .

[4]  Andrea Gasparri,et al.  On the distributed synchronization of on-line IIM interdependency models , 2009, 2009 7th IEEE International Conference on Industrial Informatics.

[5]  A. Fasano,et al.  Novel Consensus Algorithm for Wireless Sensor Networks with Noise and Interference Suppression , 2008, 2008 IEEE 10th International Symposium on Spread Spectrum Techniques and Applications.

[6]  James P. Peerenboom,et al.  Identifying, understanding, and analyzing critical infrastructure interdependencies , 2001 .

[7]  James H. Lambert,et al.  Inoperability Input-Output Model for Interdependent Infrastructure Sectors. I: Theory and Methodology , 2005 .

[8]  James H. Lambert,et al.  Inoperability Input-Output Model for Interdependent Infrastructure Sectors. II: Case Studies , 2005 .

[9]  Stefano Panzieri,et al.  Fuzzy dynamic input-output inoperability model , 2011, Int. J. Crit. Infrastructure Prot..

[10]  T. Ross.  Fuzzy Logic with Engineering Applications , 1994 .

[11]  Donato Trigiante,et al.  THEORY OF DIFFERENCE EQUATIONS Numerical Methods and Applications (Second Edition) , 2002 .

[12]  Benjamin Kuipers,et al.  Numerical Behavior Envelopes for Qualitative Models , 1993, AAAI.

[13]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[14]  Sezai Emre Tuna,et al.  Synchronizing linear systems via partial-state coupling , 2008, Autom..

[15]  Rodolphe Sepulchre,et al.  Synchronization in networks of identical linear systems , 2009, Autom..

[16]  Alfredo Germani,et al.  A New Approach to the Internally Positive Representation of Linear MIMO Systems , 2012, IEEE Transactions on Automatic Control.

[17]  V. Lakshmikantham,et al.  Theory of Fuzzy Differential Equations and Inclusions , 2003 .

[18]  C. Wu.  Synchronization in networks of nonlinear dynamical systems coupled via a directed graph , 2005 .

[19]  Sezai Emre Tuna,et al.  Conditions for Synchronizability in Arrays of Coupled Linear Systems , 2008, IEEE Transactions on Automatic Control.

[20]  Stefano Panzieri,et al.  Agent-based input-output interdependency model , 2010, Int. J. Crit. Infrastructure Prot..

[21]  Martin Hasler,et al.  Generalized connection graph method for synchronization in asymmetrical networks , 2006 .

[22]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

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

[24]  R. Varga.  Geršgorin And His Circles , 2004 .

[25]  Y. Haimes,et al.  Leontief-Based Model of Risk in Complex Interconnected Infrastructures , 2001 .

[26]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.