Quaternionic Pole Placement via Companion Forms and the Ackermann Formula
Abstract
We present an extension of state-feedback pole placement for quaternionic systems, based on companion forms and the Ackermann formula. For controllable single-input quaternionic LTI models, we define a companion polynomial that annihilates its companion matrix, characterize spectra via right-eigenvalue similarity classes, and prove coefficient-matching design in controllable coordinates. We then derive a coordinate-free Ackermann gain expression valid for real target polynomials, and state its scope and limitations. Short examples demonstrate correctness, practical use, and numerical simplicity.
Cite
@article{arxiv.2509.21425,
title = {Quaternionic Pole Placement via Companion Forms and the Ackermann Formula},
author = {Michael Sebek},
journal= {arXiv preprint arXiv:2509.21425},
year = {2026}
}
Comments
8 pages. Revised version resubmitted to IEEE Transactions on Automatic Control; proofs clarified, notation streamlined, and examples corrected. Co-funded by the European Union under the project ROBOPROX (reg. no. CZ.02.01.01/00/22_008/0004590)