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.

},
keywords = {Embedded Control, Robotics},
doi = {10.1109/TAC.2015.2480176},
url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=\&arnumber=7272075},
author = {A. Fagiolini and N. Dubbini and S. Martini and A. Bicchi}
}
@article {DM10,
title = {An equivalent of Kronecker{\textquoteright}s Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length},
journal = {Acta Arithmetica},
volume = {153},
year = {2012},
note = {accepted for publication

}, pages = {15-33}, abstract = {After defining a notion of epsilon-density, we provide for any integer m\>1 and real algebraic number alpha an estimate of the smallest epsilon such that the set of vectors of the form (t,t^alpha,...,t alpha^{m-1}) for tR is epsilon-dense modulo 1 in terms of the multiplicative Mahler measure M(A(x)) of the minimal integral polynomial A(x) of alpha, which is independent of m. In particular, we show that if alpha has degree d it is possible to take epsilon = 2^{[d/2]}/M(A(x)). On the other side we show using asymptotic estimates for Toeplitz determinants that we cannot have epsilon$-density for sufficiently large m whenever epsilon is strictly smaller than 1/M(A(x)). In the process of proving this we obtain a result of independent interest about the structure of the Z-module of integral linear recurrences of fixed length determined by a non-monic polynomial.

}, keywords = {Robotics}, issn = {0065-1036}, doi = {10.4064/aa153-1-2}, url = {http://journals.impan.pl/cgi-bin/doi?aa153-1-2}, author = {N. Dubbini and M. Monge} } @conference {DRGMP11, title = {Causality as a unifying approach between activation and connectivity analysis of fMRI data}, booktitle = {17-th annual meeting of the Organization for Human Brain Mapping}, year = {2011}, note = {Abstract}, author = {N. Dubbini and E. Ricciardi and A. Gaglianese and S. Marmi and P. Pietrini} } @article {DPB11, title = {Left invertibility of discrete-time output-quantized systems: the linear case with finite inputs}, journal = {Mathematics of Control, Signals, and Systems}, volume = {23}, number = {1 - 2}, year = {2011}, note = {published online 8 Sept. 2011

}, month = {November, 16}, pages = {117 - 139}, abstract = {This paper studies left invertibility of discrete-time linear outputquantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a nite alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility (so also left invertibility for a full measure matrix set) for all linear systems with no eigenvalue of modulus one. Therefore we are able to check left invertibility of output-quantized linear systems for a full measure matrices set. Some examples are presented to show the application of the proposed method.

}, keywords = {Robotics}, doi = {10.1007/s00498-011-0063-x}, author = {N. Dubbini and B. Piccoli and A. Bicchi} } @conference {DMB11, title = {Left invertibility of output-quantized MISO linear systems}, booktitle = {2011 Congress of the International Federation of Automatic Control - IFAC2011}, year = {2011}, month = {August 28 - Sept}, pages = {11278 - 11283}, address = {Milano, italy}, abstract = {This paper studies left invertibility of single-output discrete-time quantized linear systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility deals with the possibility of recovering unknown inputs from the only knowledge of the outputs. It is reduced, under suitable conditions, to left D-invertibility: while left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is easily (and algorithmically) detectable. Our main result is a sufficient condition for the equivalence between left invertibility and left D-invertibility in MISO system. In unidimensional systems the equivalence is valid except at most a finite (and computable) number of cases. These results allows the effective detection of left invertibility by means of left Dinvertibility, which is algorithmically detectable. An example with effective computations is presented to show the application of the proposed method.

}, keywords = {Embedded Control, Robotics}, author = {N. Dubbini and M. Monge and A. Bicchi} } @article {DPB10, title = {Left invertibility of discrete systems with finite inputs and quantized output}, journal = {International Journal Of Control}, volume = {83}, number = {4}, year = {2010}, pages = {798 - 809}, abstract = {The aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems which evolve in discrete time within a continuous state-space; quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. Our main results are obtained under some contractivity hypotheses. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system. Our main result is a necessary and sufficient condition for left invertibility and uniform left invertibility for joint contractive systems. In addition, an algorithm is proposed to recover inputs from output strings. A few examples are presented to illustrate the application of the proposed method.

}, keywords = {Embedded Control, Robotics}, author = {N. Dubbini and B. Piccoli and A. Bicchi} } @conference {DCB, title = {Left invertibility of output-quantized systems: an application to cryptography}, booktitle = {International Conference on Decision and Control - CDC 2010}, year = {2010}, month = {December}, pages = {4078-4083}, address = {Atlanta, USA}, abstract = {In this paper a secure communication method is proposed, based on left invertibility of output-quantized dynamical systems. The sender uses an output-quantized linear system with a feedback function to encode messages, which are sequences of inputs of the system. So left invertibility property enables the receiver to recover the messages. The secret key is formed by the system

}, keywords = {Robotics}, author = {N. Dubbini and A. Carluccio and A. Bicchi} } @conference {FMDB-NecSys09, title = {Distributed Consensus on Boolean Information}, booktitle = {1st IFAC Workshop on Estimation and Control of Networked Systems (NecSys{\textquoteright}09)}, year = {2009}, month = {September, 24 - }, pages = {72 - 77}, address = {Venice, Italy}, abstract = {In this paper we study the convergence towards consensus on information in a distributed system of agents communicating over a network. The particularity of this study is that the information on which the consensus is seeked is not represented by real numbers, rather by logical values or compact sets. Whereas the problems of allowing a network of agents to reach a consensus on logical functions of input events, and that of agreeing on set-valued information, have been separately addressed in previous work, in this paper we show that these problems can indeed be attacked in a unified way in the framework of Boolean distributed information systems. Based on a notion of contractivity for Boolean dynamical systems, a necessary and suficient condition ensuring the global convergence toward a unique equilibrium point is presented. This result can be seen as a first step toward the denition of a unified framework to uniformly address all consensus problems on Boolean algebras.

}, keywords = {Embedded Control, Robotics}, author = {A. Fagiolini and S. Martini and N. Dubbini and A. Bicchi} } @conference {DPB-CDC08, title = {Left invertibility of discrete systems with finite inputs and quantized output}, booktitle = {Proc. IEEE Int. Conf. on Decision and Control}, year = {2008}, pages = {4687 - 4692}, abstract = {The aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems that evolve in discrete time within a continuous state-space. Quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. We restrict to the case of contractive dynamics for fixed inputs. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the left invertibility property of the system. Our main results are a necessary and sufficient condition for a given system to be left invertible with probability one on the space of inputs (i.e. for almost all input sequences), and necessary and sufficient conditions for left invertibility and uniform left invertibility under some weak additional hypotheses. A few examples are presented to illustrate the application of the proposed method.

}, keywords = {Embedded Control, Robotics}, author = {N. Dubbini and B. Piccoli and A. Bicchi} }