Computing the Kalman form

TL;DR

This paper introduces two algorithms for computing the Kalman form of linear control systems, significantly improving algebraic complexity and practical efficiency over previous methods.

Contribution

It presents a novel algorithm based on Keller-Gehrig's technique and a practical cubic algorithm for computing the Kalman form, advancing computational efficiency.

Findings

Logarithmic matrix multiplication complexity algorithm

Cubic algorithm proven to be more efficient in practice

Significant reduction in algebraic complexity compared to prior methods

Abstract

We present two algorithms for the computation of the Kalman form of a linear control system. The first one is based on the technique developed by Keller-Gehrig for the computation of the characteristic polynomial. The cost is a logarithmic number of matrix multiplications. To our knowledge, this improves the best previously known algebraic complexity by an order of magnitude. Then we also present a cubic algorithm proven to more efficient in practice.