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Tesi etd-05262010-170955


Thesis type
Tesi di dottorato di ricerca
Author
DUBBINI, NEVIO
email address
dubbini@mail.dm.unipi.it, nevio.dubbini@gmail.com
URN
etd-05262010-170955
Title
Left invertibility of I/O quantized systems: IFS, number theory, cryptography, algorithms
Settore scientifico disciplinare
ING-INF/04
Corso di studi
MATEMATICA
Commissione
tutor Prof. Bicchi, Antonio
correlatore Piccoli, Benedetto
Parole chiave
  • Kronecker's theorem
  • left invertibility
  • IFS theory
  • Mahler measure
  • density of fractional parts
  • quantized systems
Data inizio appello
03/06/2010;
Consultabilità
parziale
Data di rilascio
03/06/2050
Riassunto analitico
This thesis addresses left invertibility for dynamical discrete--time control systems evolving in a continuous state--space, with inputs and outputs in discrete sets. More precisely, inputs are arbitrary sequences of symbols in a finite alphabet, each symbol being associated to a specific action on the system. Information available on the system is represented by<br>sequences of output values in a discrete set. Such outputs are obtained by quantization, i.e. are generated by the system evolution according to a given partition of the state-space. Left<br>invertibility has to do with the injectivity of I/O map, that is the possibility of recovering the unknown inputs that generate the observed output sequences.<br><br>Two class of systems are examined: nonlinear systems with a contraction property (called joint contraction), and linear I/O quantized systems with uniform output generating partition (with no<br>contraction hypothesis). For joint contractive systems left invertibility is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals: the IFS<br>naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system.<br>Our main result here is a necessary and sufficient condition for left invertibility. For linear I/O quantized systems left invertibility is reduced, under suitable conditions, to the notion of left D-invertibility: these conditions have to do with density of fractional parts in the unit cube, and are found using number<br>theoretic methods, involving a generalization of the classical theorem of Kronecker in terms of Mahler measure of minimal polynomials. While left invertibility takes into account membership<br>to sets of a given partition, left D-invertibility considers only membership to a single set, and is therefore much easier to<br>detect: one important result here is a method to compute left D-invertibility for all linear systems with no eigenvalue of modulus one. The main results for linear I/O quantized systems show the equivalence between left D-invertibility and left invertibility for a full measure set of matrices, and stronger results in this<br>direction are proved for unidimensional system, and for multi-input single-output (MISO) systems.<br><br>Moreover a cryptographic system based on left invertibility of I/O quantized systems is presented, which provides a secure<br>communication method. Finally algorithms are given, to check left invertibility of a I/O quantized linear systems, and to recover<br>inputs from the knowledge of the output sequences.
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