Quaternionic Pole Placement via Companion Forms and the Ackermann Formula
This provides a theoretical extension for quaternionic control systems, which is incremental to existing real/complex methods.
The authors tackled the problem of state-feedback pole placement for quaternionic linear time-invariant systems by extending companion forms and the Ackermann formula, proving coefficient-matching design in controllable coordinates and deriving a coordinate-free gain expression for real target polynomials.
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.