Quaternionic Pole Placement via Companion Forms and the Ackermann Formula

1. Problem Statement

The paper addresses the fundamental control problem of state-feedback pole placement for single-input, controllable, linear time-invariant (LTI) systems defined over the field of quaternions ($\mathbb{H}$).

• Context: While pole placement is a standard technique in real ($\mathbb{R}$) and complex ($\mathbb{C}$) systems, extending it to quaternions is non-trivial due to the non-commutative nature of quaternion multiplication.
• Key Challenges:
  • Eigenvalue Definition: Right eigenvalues ($Av = v\lambda$) are defined only up to quaternionic similarity classes ($\lambda \sim \alpha^{-1}\lambda\alpha$). A single eigenvalue represents a "sphere" of equivalent values, not a unique point.
  • Lack of Determinants: Standard tools like the characteristic polynomial, Cayley-Hamilton theorem, and adjugate matrices do not generalize cleanly to quaternions because determinants are ill-defined or lack necessary properties in non-commutative algebras.
  • Evaluation Order: Polynomial evaluation is order-dependent (left vs. right), meaning $p(A)$ is not unique unless coefficients are central (real).
• Goal: To develop a rigorous, determinant-free framework for assigning the closed-loop right spectrum (a set of similarity classes) using state feedback $u = -Kx$.

2. Methodology

The author proposes a two-pronged approach that mirrors classical $\mathbb{R}/\mathbb{C}$ workflows but adapts them to the algebraic constraints of $\mathbb{H}$.

A. Quaternionic Companion Forms

Instead of relying on determinants to find a characteristic polynomial, the paper constructs a controllable companion form directly from the controllability matrix.

• Construction: For a controllable pair $(A, B)$, a similarity transformation $T$ is derived using the inverse of the controllability matrix $C = [B, AB, \dots, A^{n-1}B]$.
• Result: The system is transformed into a lower companion form $(A_c, B_c)$ where $A_c$ has a specific structure (shift matrix with the last row containing coefficients).
• Companion Polynomial: A monic quaternionic polynomial $a(\lambda)$ is defined based on the last row of $A_c$. Crucially, the paper proves that this polynomial right-annihilates the companion matrix: $a(A_c)_\mathcal{R} = 0$.
• Spectral Characterization: The right spectrum of $A_c$ (and thus the original $A$) is exactly the set of similarity classes of the right zeros of $a(\lambda)$.

B. Pole Placement Strategies

The paper derives two equivalent design methods to assign a desired closed-loop spectrum:

1. Coefficient Matching (General Case):

   • Transform the system to the companion form.
   • Select a desired monic quaternionic polynomial $a_d(\lambda)$ whose right zeros represent the target similarity classes.
   • Compute the feedback gain $K_c$ in companion coordinates by matching coefficients: $K_c = [d_0 - a_0, \dots, d_{n-1} - a_{n-1}]$.
   • Transform back to original coordinates: $K = K_c T^{-1}$.
   • Scope: Valid for any quaternionic target polynomial (including non-real coefficients).

2. Quaternionic Ackermann Formula (Restricted Case):

   • Derives a coordinate-free expression for the gain $K$ that avoids explicit transformation to companion coordinates.
   • Constraint: The desired polynomial $a_d(\lambda)$ must have real (central) coefficients ($a_d \in \mathbb{R}[\lambda]$). This is necessary because real scalars commute with all quaternions, ensuring the evaluation $a_d(A)$ is unambiguous and satisfies the "intertwining" property required for the proof.
   • Formula: $K = e_n^T C^{-1} a_d(A)$, where $e_n$ is the last canonical basis vector.
   • Scope: Covers real poles and complex-conjugate pairs (which define spherical classes in $\mathbb{H}$), but fails for arbitrary quaternionic coefficients.

3. Key Contributions

• Determinant-Free Framework: The paper establishes a complete theory for pole placement without using determinants, Cayley-Hamilton, or adjugates, which are problematic in $\mathbb{H}$.
• Companion Polynomial Theory: It rigorously defines the companion polynomial for controllable pairs and proves it right-annihilates the companion matrix, serving as the functional equivalent of the characteristic polynomial in this context.
• Generalized Ackermann Formula: It provides the first explicit Ackermann-type gain formula for quaternionic systems, clearly delineating its validity (real coefficients) and limitations.
• Spectral Interpretation: It clarifies that pole placement in $\mathbb{H}$ targets similarity classes of right eigenvalues, not individual points, and shows how real/complex polynomials encode these classes.

4. Results and Validation

• Theoretical Proofs: Theorems 1–6 provide the mathematical foundation, proving the existence of the companion form, the annihilation property, the spectral correspondence, and the correctness of the Ackermann formula under real-coefficient constraints.
• Numerical Examples: The paper includes examples demonstrating:
  • The calculation of the transformation matrix $T$ and companion form.
  • Successful pole placement for real targets (e.g., $\lambda = -1, -2$).
  • Successful pole placement for complex conjugate pairs (e.g., $\lambda = -1 \pm i$), which correspond to a single spherical class in $\mathbb{H}$.
  • Failure Case: A demonstration showing that applying the Ackermann formula to a polynomial with non-real (quaternionic) coefficients yields an incorrect gain and fails to assign the desired spectrum.

5. Significance

This work bridges a critical gap in quaternionic control theory. By providing practical, computable design tools (companion forms and Ackermann formula) that respect the non-commutative algebra of quaternions, it enables the direct application of state-feedback control to systems modeled in hypercomplex space.

• Applications: This is particularly relevant for rigid-body attitude control, quantum computing (qubit gate control), NMR/MRI pulse design, and color image processing, where quaternionic models are natural but previously lacked robust control synthesis tools.
• Practicality: The methods retain the familiar workflow of classical control (companion coordinates, coefficient matching) while explicitly handling the nuances of quaternionic similarity and evaluation order, making them accessible to engineers and researchers.

In summary, Sebek successfully lifts standard state-space pole placement techniques to the quaternionic domain, offering a robust, determinant-free methodology that is both theoretically sound and practically applicable to single-input controllable systems.