Observer Design over Hypercomplex Quaternions

TL;DR

The paper addresses observer design for dynamical systems over the quaternion algebra $\mathbb{H}$, where noncommutativity invalidates determinant- and polynomial-based methods. It develops a determinant-free, characteristic-polynomial-free framework built around a right observable companion form and a quaternionic Ackermann formula to place right-eigenvalue similarity classes using real-coefficient polynomials. By working directly over $\mathbb{H}$, the approach preserves the noncommutative structure and clarifies the role of the right spectrum and similarity classes, yielding straightforward full-order observer recipes directly over quaternions. Numerical examples illustrate advantages over vectorized or complex-adjoint surrogates and highlight when classical one-shot formulas remain valid.

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 its companion polynomial that encodes error dynamics via right-eigenvalue similarity classes. The design mirrors the real/complex case - coefficient updates in companion coordinates, followed by a similarity back - yet avoids determinants, characteristic/minimal polynomials, and Cayley-Hamilton identities that do not transfer to quaternions. We also give an Ackermann-type construction for the important case of closed-loop companion polynomials with real coefficients, ensuring similarity-equivariant evaluation. The results yield simple recipes for full-order observers directly over quaternions, clarify the role of right spectra and their similarity classes, and pinpoint when classical one-shot formulas remain valid. Numerical examples illustrate the method and advantages over vectorized or complex-adjoint surrogates.