# On recursive computation of coprime factorizations of rational matrices

## TL;DR

This paper introduces recursive pole dislocation methods for computing minimal-degree coprime factorizations of rational matrices, ensuring stability and properness through descriptor state-space realizations.

## Contribution

It presents novel recursive pole dislocation techniques that enable minimal-degree coprime factorizations with stable and proper factors, applicable to improper rational matrices.

## Key findings

- Successfully computes coprime factorizations with minimal denominator degree.
- Ensures numerical reliability in pole placement and factorization.
- Provides implementation details and illustrative examples.

## Abstract

General computational methods based on descriptor state-space realizations are proposed to compute coprime factorizations of rational matrices with minimum degree denominators. The new methods rely on recursive pole dislocation techniques, which allow to successively place all poles of the factors into a "good" region of the complex plane. The resulting McMillan degree of the denominator factor is equal to the number of poles lying in the complementary "bad" region and therefore is minimal. The developed pole dislocation techniques are instrumental for devising numerically reliable procedures for the computation of coprime factorizations with proper and stable factors of arbitrary improper rational matrices and coprime factorizations with inner denominators. Implementation aspects of the proposed algorithms are discussed and illustrative examples are given.