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Quaternionic Pole Placement via Companion Forms and the Ackermann Formula

Michael Sebek

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 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.

Quaternionic Pole Placement via Companion Forms and the Ackermann Formula

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 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.

Paper Structure

This paper contains 8 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Quaternions and Matrices
  3. Quaternionic state–space systems
  4. Companion Form and Companion Polynomial
  5. Control Problem: State-Feedback Pole Placement in Quaternionic Systems
  6. Quaternionic Pole Placement via Controllable Companion Form
  7. Quaternionic Ackermann Formula
  8. Conclusion

Key Result

Theorem 1

Let $A \in \mathbb{H}^{n\times n}$ and $B \in \mathbb{H}^{n}$ be given, and assume the controllability matrix $\mathcal{C}=[B,AB,\dots,A^{n-1}B]$ is invertible over $\mathbb{H}$.

Theorems & Definitions (27)

  • Definition 1: Controllable companion form and companion polynomial
  • Theorem 1: Determinant-free companion form for a controllable pair
  • Remark 1
  • proof
  • Example 1
  • Theorem 2: Companion polynomial annihilates its companion matrix over $\mathbb{H}$
  • proof
  • Remark 2
  • Theorem 3: Right zeros form the right spectrum of $A_c$
  • proof
  • ...and 17 more