Observer Design over Hypercomplex Quaternions

Abstract

We develop observer design over hypercomplex quaternions in a characteristic-polynomial-free framework. Using the standard right-module convention, we derive a right observable companion form and companion polynomial that encode error dynamics through right-eigenvalue similarity classes. We also give an Ackermann-type formula for real-coefficient target polynomials, where polynomial evaluation is similarity-equivariant. The resulting recipes place observer poles directly over quaternions and clarify when companion-coordinate updates and one-shot Ackermann formulas remain valid.

Accepted for presentation at the 24th European Control Conference (ECC 2026), Reykjavik, Iceland. This work was co-funded by the European Union under the project ROBOPROX (reg. no. CZ.02.01.01/00/22 008/0004590)