Left invertibility of discrete-time output-quantized systems: the linear case with finite inputs

TL;DR

This paper investigates the conditions under which discrete-time linear output-quantized systems are left invertible, providing a method to verify invertibility for systems with no eigenvalues of modulus one, applicable to a broad class of matrices.

Contribution

It introduces a new approach to determine left invertibility of output-quantized linear systems using left D-invertibility, especially for systems with algebraically independent dynamic matrices.

Findings

Left invertibility reduces to left D-invertibility under certain conditions.

The method applies to all systems with no eigenvalues of modulus one.

Invertibility can be checked for a full measure set of matrices.

Abstract

This paper studies left invertibility of discrete-time linear output-quantized systems. 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 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 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 set of matrices. Some examples are presented to show the application of the proposed method.