⚡ electrical engineering

Observer Design over Hypercomplex Quaternions

This paper presents a characteristic-polynomial-free framework for observer design over hypercomplex quaternions, deriving a right observable companion form and an Ackermann-type formula that enable direct pole placement via right-eigenvalue similarity classes while clarifying the validity conditions for companion-coordinate updates.

Original authors: Michael Sebek

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Sebek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to navigate a spaceship through a stormy nebula. You have a map (the system model) and a sensor (the output), but your sensors are a bit noisy. You need a "co-pilot" (an observer) that can guess your exact location and speed based on the noisy sensor data, so you can steer the ship correctly.

In the world of standard math (using real numbers or complex numbers), we have a well-worn playbook for building this co-pilot. We use tools like "determinants" and "characteristic polynomials" to calculate exactly where the co-pilot should be.

The Problem:
This paper deals with Quaternions. Think of quaternions as a super-charged version of numbers used to describe 3D rotations (like how a drone or a spaceship spins).

The Catch: In normal math, $3 \times 4$ is the same as $4 \times 3$ . In the world of Quaternions, order matters. $A \times B$ is often not the same as $B \times A$ .
The Consequence: Because of this "non-commutative" nature, all the standard math tools (like the characteristic polynomial) break. They are like trying to use a screwdriver to hammer a nail; they just don't fit. You can't just plug the numbers into the old formulas and expect them to work.

The Solution:
Michael Sebek, the author, has built a new, custom toolkit specifically for Quaternions. He didn't try to force the old tools to work; he invented a new way to build the co-pilot that respects the unique rules of Quaternion math.

Here is how the paper works, explained through analogies:

1. The "Right-Handed" Rule

In normal math, you can multiply numbers from the left or the right, and it's usually fine. In Quaternions, you have to be very strict about which side you multiply on.

The Paper's Approach: The author decides to stick strictly to a "Right-Module" convention. Imagine everyone in the spaceship is right-handed. If you try to shake hands with your left hand, things get messy. By forcing everyone to use their right hand (multiplying on the right), the math becomes consistent and predictable again.

2. The "Companion Form" (The Blueprint)

Usually, to design a controller, you calculate a "Characteristic Polynomial" (a complex equation that describes the system's behavior).

The Old Way: Calculate the equation first, then build the machine.
The New Way (This Paper): Since the "equation" tool is broken for Quaternions, the author skips it entirely. Instead, he builds a standard blueprint called the "Observable Companion Form."
The Analogy: Imagine you want to build a house. Instead of calculating the theoretical stress limits of every beam first (which is hard/impossible here), you just build the house using a specific, pre-approved architectural style (the Companion Form). Once the house is built, you can read the properties directly off the walls. You don't need the complex equation to know how the house behaves; the structure itself tells you.

3. The "Similarity Classes" (The Family Reunion)

In normal math, if a system has a "pole" (a number that determines stability) at $-2$, it stays at $-2$.

The Quaternion Twist: In Quaternions, numbers can be "similar" even if they look different. For example, $-2 + 3i$ and $-2 + 3j$ might belong to the same "family" or Similarity Class. They act the same way in the system, even though they look different.
The Goal: The author's method doesn't just place a single number; it places an entire family of numbers. It ensures the co-pilot is stable no matter which "member" of the family the system happens to be in.

4. The "Ackermann Formula" (The Shortcut)

Engineers love shortcuts. There is a famous formula called Ackermann's Formula that lets you calculate the co-pilot's settings in one single step.

The Limitation: The author found that this shortcut only works if your target settings are "Real" numbers (like $-2$ or $-5$).
The Discovery: If you want to target a complex Quaternion number (like $-2 + 3j$ ), the shortcut fails. It gives you the wrong answer.
The Fix: You have to use the "Blueprint" method (updating the coefficients in the Companion Form) to get the right answer for complex targets. The paper clarifies exactly when you can use the fast shortcut and when you must do the full calculation.

Why Does This Matter?

You might ask, "Who cares about Quaternions?"

Robotics & Drones: They spin in 3D space. Quaternions are the best way to describe that spin without getting "gimbal locked" (a math error where you lose control).
Medical Imaging (MRI): The signals in MRI machines are often processed using Quaternions to handle color and polarization.
Quantum Computing: The math of quantum mechanics is deeply rooted in these structures.

The Bottom Line:
This paper is like a mechanic who realized the standard wrench set doesn't fit the bolts on a new type of alien engine. Instead of forcing the wrenches, they designed a new set of tools that fit perfectly. They showed us:

How to build a stable observer (co-pilot) for Quaternion systems without using broken tools.
How to read the system's behavior directly from a special structure.
Exactly when you can use the "fast track" (Ackermann's formula) and when you must take the "scenic route" (Companion Form).

This allows engineers to design better, more stable control systems for drones, robots, and quantum computers, keeping them native to the Quaternion language rather than clumsily translating them into standard math first.

1. Problem Statement

The paper addresses the challenge of designing state observers for linear time-invariant (LTI) systems defined over the division algebra of quaternions ( $\mathbb{H}$ ).

The Core Difficulty: Unlike real ( $\mathbb{R}$ $R$ ) or complex ( $\mathbb{C}$ $C$ ) numbers, quaternion multiplication is non-commutative. This fundamental property breaks standard linear algebra tools:
- Determinants and characteristic polynomials do not generalize straightforwardly.
- The Cayley-Hamilton theorem fails in its classical form.
- Eigenvalues are not unique scalars but exist as similarity classes ( $[\lambda] = \{\alpha^{-1}\lambda\alpha \mid \alpha \in \mathbb{H} \setminus \{0\}\}$ ).
- Polynomial evaluation depends on whether coefficients act from the left or right.
Current Limitations: Existing applications (e.g., in aerospace and robotics) typically convert quaternionic systems into real 4-vector coordinates to use standard observers. This approach loses the intrinsic hypercomplex structure and prevents reasoning directly in terms of quaternionic spectra and similarity classes.
Goal: Develop a native observer design framework over $\mathbb{H}$ that avoids determinants and characteristic polynomials, directly assigning observer poles as quaternionic similarity classes.

2. Methodology

The author proposes a determinant-free, characteristic-polynomial-free framework based on observability and duality.

A. Mathematical Foundations

Right-Module Convention: The system is modeled as a right $\mathbb{H}$ -module where matrices act from the left and scalars multiply from the right ($Ax + Bu$).
Spectrum Definition: Stability and pole placement are defined via right eigenvalues ( $Av = v\lambda$ ), which are invariant under similarity transformations.
Duality: The paper establishes that the correct dual of a system $(A, C)$ is $(A^*, C^*)$ , where $*$ denotes the conjugate transpose. This preserves the Hermitian form and spectral properties, unlike simple transposition.

B. Observable Companion Form

Instead of using a characteristic polynomial, the paper constructs a Right Observable Companion Form directly from the observability matrix $\mathcal{O}$ :

Construction: Given an observable pair $(A, C)$ , a similarity transformation $T$ is derived using the inverse of the observability matrix ( $T = [\mathcal{O}^{-1}e_n, A\mathcal{O}^{-1}e_n, \dots]$ ).
Result: The system is transformed into a canonical form $(A_o, C_o)$ where $A_o$ has a specific subdiagonal structure and the last column contains the coefficients of a Companion Polynomial $a(\lambda)$ .
Key Property: The paper proves that while the classical Cayley-Hamilton theorem fails, the companion polynomial annihilates the companion matrix $A_o$ under left substitution ( $a(A_o)_\ell = 0$ ).

C. Pole Placement Strategies

The paper presents two methods to assign observer poles (right-eigenvalue similarity classes):

Companion Coordinate Update:
- Transform the system to observable companion form.
- Calculate the difference between the desired polynomial coefficients ( $d_i$ ) and the current coefficients ( $a_i$ ).
- Update the observer gain directly in the transformed coordinates and map back to the original system.
- Capability: This method works for any target polynomial, including those with non-real (non-central) quaternionic coefficients.
Quaternionic Ackermann's Formula:
- A direct, one-shot formula for the observer gain $L$ :
  $L = a_d(A) \mathcal{O}^{-1} e_n$
- Constraint: This formula is valid only if the desired polynomial $a_d(\lambda)$ has real coefficients. Real coefficients lie in the center of $\mathbb{H}$ , ensuring that polynomial evaluation is similarity-equivariant. If the target polynomial has quaternionic coefficients, this formula yields incorrect results.

3. Key Contributions

Native Quaternionic Observer Design: The first derivation of a full observer design framework that operates directly over $\mathbb{H}$ without converting to real coordinates.
Determinant-Free Framework: Replaces non-existent or problematic quaternionic characteristic polynomials with a constructive Observable Companion Form derived purely from the observability matrix.
Spectral Characterization: Proves that the right spectrum of the observable companion matrix corresponds exactly to the similarity classes of the right zeros of its companion polynomial.
Refined Ackermann Formula: Derives a quaternionic Observer Ackermann formula that is rigorously proven to work only for real-coefficient target polynomials, clarifying a common misconception about applying classical formulas to non-commutative algebras.
Duality Clarification: Explicitly defines the controller-observer duality over $\mathbb{H}$ using the conjugate transpose ( $*$ ) rather than the standard transpose.

4. Results

Theoretical Validation: Theorems 1–6 provide rigorous proofs for the existence of the companion form, the annihilation property of the companion polynomial, and the correctness of the pole placement methods.
Simulation: A simulation example (Example 5) demonstrates the convergence of a quaternionic observer.
- System: 2nd-order SISO system with quaternionic matrices.
- Outcome: The estimated state $\hat{x}(t)$ converges to the true state $x(t)$ in all four components (scalar, $i$ , $j$ , $k$ ) when using the derived gains.
Comparative Analysis: The paper demonstrates that while the Companion Coordinate Update method successfully handles quaternionic target poles (e.g., $-1+j$ ), the Ackermann Formula fails in this specific case, reinforcing the necessity of the real-coefficient restriction for the shortcut formula.

5. Significance

Structural Preservation: This work allows engineers and researchers to design filters and observers that respect the intrinsic geometric and algebraic structure of quaternionic data (crucial for 3D rotations, polarization, and quantum dynamics).
Algorithmic Efficiency: By avoiding the conversion to 4D real vectors, the proposed methods can be more computationally efficient and numerically stable for high-dimensional quaternionic systems.
Theoretical Clarity: It resolves ambiguities regarding eigenvalues and polynomial evaluation in non-commutative settings, providing a clear "recipe" for when standard control intuition holds and when it requires modification (specifically regarding the side of multiplication and coefficient centrality).
Application Scope: The framework is directly applicable to fields such as quantum control, NMR/MRI pulse design, color imaging, and quaternion-valued neural networks, where maintaining the hypercomplex structure is algorithmically meaningful.