Numerically Reliable Brunovsky Transformations

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.