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arXiv · 2509.21425

Quaternionic Pole Placement via Companion Forms and the Ackermann Formula

Michael Sebek
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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.

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Michael Sebek. 2026-06-14. Quaternionic Pole Placement via Companion Forms and the Ackermann Formula. https://doi.org/10.1109/tac.2026.3702917

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