Exploring Noncommutative Polynomial Equation Methods for Discrete-Time Quaternionic Control

TL;DR

This paper introduces novel polynomial methods for discrete-time quaternionic control systems, adapting classical control techniques to handle noncommutative quaternion algebra and proposing a quaternionic pole placement approach.

Contribution

It develops quaternionic polynomial-based control methods, including transfer function representations and a new feedback design procedure for quaternionic pole placement.

Findings

Right zeros relate to similarity classes of quaternionic eigenvalues.

A new feedback design procedure generalizes pole placement to quaternionic systems.

Quaternionic polynomial equations are effectively used for control design.

Abstract

We present new polynomial-based methods for discrete-time quaternionic systems, highlighting how noncommutative multiplication modifies classical control approaches. Defining quaternionic polynomials via a backward-shift operator, we examine left and right fraction representations of transfer functions, showing that right zeros correspond to similarity classes of quaternionic matrix right eigenvalues. We then propose a feedback design procedure that generalizes pole placement to quaternions - a first approach using a genuine quaternionic polynomial equation.