Quaternionic Pole Placement via Companion Forms and the Ackermann Formula

2025/09/25 by Sebek, Michael · 1 citation
#15A21 #15A66 #93B55 #93C05 #FOS: Electrical engineering #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2509.21425

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 right-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.

Citations

• Quaternions in Signal and Image Processing: A comprehensive and objective overview
• Quaternion Convolutional Neural Networks for End-to-End Automatic Speech\n Recognition
• Standard Polynomial Equations over Division Algebras

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