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

TL;DR

This paper investigates the conditions under which one-dimensional discrete-time linear systems with quantized outputs are invertible, showing that left invertibility and a simpler form called left D-invertibility are equivalent for most system parameters, using number theory.

Contribution

It establishes the equivalence of left invertibility and left D-invertibility for almost all system parameters in 1D quantized linear systems, employing number theoretic methods.

Findings

Left invertibility and left D-invertibility are equivalent for all but a finite set of parameters.

The equivalence holds except possibly at two accumulation points of parameter values.

Number theory techniques are used to identify the exceptional parameter set.

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.