# Generalized algorithms for the approximate matrix polynomial GCD of reducing data uncertainties with application to MIMO system and control

## TL;DR

This paper extends algorithms for approximate polynomial GCD to matrix polynomials, enabling better handling of data uncertainties in control systems and signal processing applications.

## Contribution

It generalizes two scalar polynomial GCD algorithms to matrix polynomials, including a fast and a more accurate method, with application to MIMO systems.

## Key findings

- Both algorithms perform similarly to scalar cases.
- The methods effectively handle data uncertainties.
- Application demonstrated on MIMO control systems.

## Abstract

Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific literature in the framework of matrix polynomials seems to be limited to the problem of exact greatest common divisors computation. In this paper, we generalize two algorithms from scalar to matrix polynomials. The first one is fast and simple. The second one is more accurate but computationally more expensive. We test the performances of the two algorithms and observe similar behavior to the one in the scalar case. Finally we describe an application to multi-input multi-output linear time-invariant dynamical systems.