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arXiv · 0911.0768

Left invertibility of I/O quantized linear systems in dimension 1: a number theoretic approach

Nevio Dubbini·Maurizio Monge·Antonio Bicchi
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Abstract

This paper studies left invertibility of discrete-time linear I/O quantized linear systems of dimension 1. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite 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 in sets of a given partition, left D-invertibility considers only distances, and is very easy to detect. Considering the system $x^+=ax+u$, our main result states that left invertibility and left D-invertibility are equivalent, for all but a (computable) set of $a$'s, discrete except for the possible presence of two accumulation point. In other words, from a practical point of view left invertibility and left D--invertibility are equivalent except for a finite number of cases. The proof of this equivalence involves some number theoretic techniques that have revealed a mathematical problem important in itself. Finally, some examples are presented to show the application of the proposed method.

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Nevio Dubbini, Maurizio Monge, Antonio Bicchi. 2009-11-04. Left invertibility of I/O quantized linear systems in dimension 1: a number theoretic approach. https://arxiv.org/abs/0911.0768

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