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.

8 pages. Accepted manuscript / author final version. The IEEE Early Access version is available in IEEE Xplore with DOI 10.1109/TAC.2026.3702917; the article is accepted for publication in a future issue of IEEE Transactions on Automatic Control and may still be copy-edited. Co-funded by the European Union under the project ROBOPROX (reg. no. CZ.02.01.01/00/22_008/0004590)