⚡ electrical engineering

An Algebraic State Observer for a Self-Sensing Active Magnetic Bearing System

This paper presents a novel globally stable algebraic state observer for self-sensing active magnetic bearing systems that utilizes only current and voltage measurements to establish an algebraic relationship between unmeasurable states and filtered inputs/outputs, subsequently extending this design into a robust asymptotic observer validated by simulations.

Original authors: Olga Zarina, Natalya Martyuhova, Alexey Bobtsov, Romeo Ortega

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Olga Zarina, Natalya Martyuhova, Alexey Bobtsov, Romeo Ortega

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

The Big Picture: The "Invisible" Rotor

Imagine a high-tech spinning machine, like a turbine or a generator, that floats in mid-air without touching anything. This is an Active Magnetic Bearing (AMB). Instead of oil or grease, powerful magnets hold the spinning part (the rotor) in place.

Usually, to keep this rotor from crashing into the walls, the computer needs to know exactly where it is. To do this, traditional systems use physical sensors (like a camera or a ruler) to measure the rotor's position.

The Problem: These physical sensors are fragile, expensive, and can't survive extreme environments (like a vacuum or super-high heat). The goal of this paper is to remove these sensors entirely.

The Challenge: If you remove the sensor, how does the computer know where the rotor is? It only has access to two things:

The voltage it sends to the magnets (the "push").
The current flowing through the magnets (the "effort").

The researchers wanted to build a mathematical "detective" (an observer) that looks at the voltage and current and figures out the hidden position and speed of the rotor, without ever needing a physical sensor.

The Solution: Two Types of "Mathematical Detectives"

The paper proposes two different ways to solve this mystery. Think of them as two different strategies for guessing a secret number.

1. The "Instant Calculator" (The Algebraic Observer)

The researchers first tried to create a "perfect" detective. They wanted a formula that could take the current voltage and current, do a specific math calculation, and immediately spit out the exact position of the rotor.

The Analogy: Imagine you are trying to guess the weight of a mystery box. You have a formula that says: "If I know the exact force you used to lift it and the exact speed it moved, I can calculate the weight instantly."
The Catch: In the real world, the rotor often sits still or moves very smoothly. This is like trying to guess the weight of a box that isn't moving at all. The math requires the box to be "wiggling" a bit (excitation) to work. If the rotor is too calm, the formula hits a "division by zero" error and breaks. It's like trying to solve a puzzle where a crucial piece is missing.

2. The "Patient Learner" (The Asymptotic Observer)

Since the "Instant Calculator" sometimes fails when the machine is too quiet, the researchers built a second detective: the Asymptotic Observer.

The Analogy: Instead of demanding the answer instantly, this detective says, "I might not know the exact position right this second, but I will watch the voltage and current closely. Over time, as I see more data, my guess will get closer and closer to the truth until it is perfect."
How it works: It starts with a guess and constantly corrects itself. It doesn't need the machine to be "wiggling" as much as the first one. It avoids the math errors (singularities) that broke the first detective.

How They Tested It

The researchers didn't build a physical machine for this paper; they built a computer simulation. They created a virtual magnetic bearing and fed it different types of electrical signals:

Constant Push: They gave the magnets a steady, unchanging voltage.
- Result: The "Patient Learner" worked, but it took a while to settle down.
Wiggly Push: They gave the magnets a vibrating, wavy voltage (like a sine wave).
- Result: This "excited" the system. The "Patient Learner" figured out the position much faster and more accurately. This proved that the system works best when the machine is moving or vibrating slightly.

The Takeaway

The paper claims to have successfully designed a new type of mathematical tool that can estimate the position of a floating magnetic rotor using only electrical signals (voltage and current), with no physical sensors needed.

They created a "perfect" instant formula, but it's fragile and breaks if the machine is too quiet.
They created a "patient" version that learns over time, which is more robust and works even when the machine is calm.

What they didn't do:

They did not build a real-world prototype in a factory or a vacuum chamber yet (they only simulated it).
They did not claim this works for medical devices or other specific industries yet; they only focused on the math for this specific type of magnetic bearing.
They admitted that while the math works, the "Instant Calculator" version still struggles when the system lacks "excitation" (movement), which is why they rely on the "Patient Learner" version.

In short, they found a clever way to "see" the invisible rotor using only electricity, offering a path toward cheaper, tougher, and sensor-free magnetic machines.

Technical Summary: An Algebraic State Observer for a Self-Sensing Active Magnetic Bearing System

Problem Formulation
This paper addresses the design of a globally stable state observer for a single-axis Active Magnetic Bearing (AMB) system operating in a self-sensing configuration. In this scenario, the system is equipped only with measurements of currents ( $y$ ) and voltage inputs ( $u$ ); displacement sensors are absent. The objective is to estimate the full state vector $x = \text{col}(\phi_1, \phi_2, q, p)$ , comprising flux linkages, rotor position, and momentum, using only these available signals and known system parameters ( $c_1, c_2, R, N$ ).

The authors distinguish their approach from standard observer design. Rather than constructing a dynamical system where the state estimate asymptotically converges to the true state, the primary goal is to establish an algebraic relation between the unmeasurable state and filtered versions of the inputs and outputs that holds true for all $t \geq 0$ .

Methodology
The proposed methodology relies on a novel application of the Swapping Lemma (Lemma 3.6.5 in [14]), a technique previously utilized in [13] for other systems. The approach proceeds in two stages:

Algebraic Swapping Lemma Observer (ASLO):
The authors derive an explicit algebraic mapping $\mathcal{O}$ such that $x(t) = \mathcal{O}[u(t), y(t), \mathcal{F}[u](t), \mathcal{F}[y](t)]$ . This derivation involves defining a Linear Time-Invariant (LTI) filter $\mathcal{F}(p) = \frac{\lambda}{p+\lambda}$ and constructing a set of measurable signals ( $z_i, w_i$ ) through dynamic extensions involving the filter and the system inputs/outputs. The final state estimate for the flux coordinates is obtained via a matrix inversion involving a scalar signal $\Delta(t) = w_3 w_6 - w_4 w_5$ .
Asymptotic Algebraic Swapping Lemma Observer (A-ASLO):
The authors identify a critical limitation in the ASLO: the requirement that $\Delta(t) \neq 0$ for the matrix inversion to be valid. In AMB applications, operating conditions often provide insufficient "excitation," leading to singularities where $\Delta(t)$ approaches zero. To overcome this, the authors propose an asymptotic version (A-ASLO).
- For the flux coordinates, the A-ASLO replaces the algebraic inversion with a dynamic estimator driven by the error between the algebraic solution and the current estimate, ensuring convergence provided $\Delta(t) \notin \mathcal{L}_2$ .
- For the position and velocity coordinates, a separate dynamic observer is constructed using the estimated fluxes, ensuring convergence under the generic assumption that the estimated flux magnitude is non-zero.

Key Contributions

First Algebraic Observer for AMBs: This work presents the first algebraic state observer specifically designed for Active Magnetic Bearing systems.
Novel Derivation Technique: The paper introduces the use of the Swapping Lemma to derive an exact algebraic relation for the state, avoiding the successive signal differentiation often found in other "algebraic" observer claims (which are noted as practically inadmissible).
Robustness to Lack of Excitation: By transitioning from the exact ASLO to the asymptotic A-ASLO, the authors address the singularity issues inherent in the algebraic approach when system excitation is low, a common condition in AMB operation.

Simulation Results
The performance of the A-ASLO was evaluated through simulations of a 2-DOF AMB system with parameters $R=1.6$ , $N=321$ , $c_1=c_2=293.5$ , and $m=2.5$ .

Constant Inputs: Under constant input signals ( $u_1=u_2=1$ ), the observer demonstrated convergence of estimation errors for flux, position, and velocity, though the transient behavior varied with gain parameters ( $\gamma_1, \gamma_2$ ) and the filter parameter $w_0$ .
Sinusoidal Inputs: When the system was excited with sinusoidal inputs ( $u_1 = 50\sin(5t), u_2 = 30\sin(7t)$ ), the simulation results showed a "beneficial effect of exciting the system," leading to improved transient performance and faster error convergence compared to the constant input case.

Significance and Claims
The paper claims to provide a fundamentally different approach to state observation in AMBs by establishing an algebraic identity for the state rather than relying solely on asymptotic convergence. However, the authors are modest regarding the immediate practicality of the pure algebraic solution (ASLO). They explicitly state that "due to the poor 'excitation' conditions" typical of AMB operation, the performance of the ASLO is "below par."

Consequently, the primary significance of the work lies in the derivation of the A-ASLO, which retains the theoretical foundation of the algebraic approach while avoiding the singularity problems that render the pure algebraic solution infeasible in low-excitation scenarios. The authors note that this singularity issue was not observed in previous applications of the same technique to other systems (e.g., PMSMs, induction motors, ball-and-beam systems), highlighting the specific challenges posed by AMB dynamics.

Future Directions
The authors outline three specific directions for future research:

Developing an adaptive version of the ASLO to estimate unknown parameters, specifically resistance ( $R$ ) and external disturbance forces ( $f_e$ ).
Conducting experimental validation of the proposed results.
Investigating methods to improve ASLO performance, potentially by utilizing alternative variations of the Swapping Lemma reported in [19].