Numerically Reliable Brunovsky Transformations

TL;DR

This paper introduces a numerically reliable method for computing Brunovsky transformations with lower errors and better conditioning, enhancing computational optimal control applications.

Contribution

It presents a new technique combining staircase form reduction, linear parametrization, and optimization to improve the stability and accuracy of Brunovsky transformations.

Findings

Lower construction errors compared to existing methods

Improved conditioning of the transformations

Enhanced numerical stability through optimization

Abstract

The Brunovsky canonical form provides sparse structural representations that are beneficial for computational optimal control, yet existing methods fail to compute it reliably. We propose a technique that produces Brunovsky transformations with substantially lower construction errors and improved conditioning. A controllable linear system is first reduced to the staircase form via an orthogonal similarity transformation. We then derive a simple linear parametrization of the transformations yielding the unique Brunovsky form. Numerical stability is further enhanced by applying a deadbeat gain before computing system matrix powers and by optimizing the linear parameters to minimize condition numbers.