# MRAS-based sensorless field oriented controlled axial flux permanent magnet synchronous machine with additional parameter adaptation

Michael Brüns, Christan Rudolph

► To cite this version:

Michael Brüns, Christan Rudolph. MRAS-based sensorless field oriented controlled axial flux permanent magnet synchronous machine with additional parameter adaptation. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0348. hal-05067875

HAL Id: hal-05067875

https://utc.hal.science/hal-05067875v1

Submitted on 14 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# MRAS-based sensorless field oriented controlled axial flux permanent magnet synchronous machine with additional parameter adaptation

Michael Brüns, Christan Rudolph

Hamburg University of Applied Sciences

Faculty of Engineering and Computer Science

Berliner Tor 21

Hamburg, Germany

E-Mail: Michael.Bruens@haw-hamburg.de; Christian.Rudolph@haw-hamburg.de;

URL: www.haw-hamburg.de

## Keywords

Sensorless control, Adaptive control, Stability analysis, Axial machines, Non-linear control.

## Abstract

The characteristics of a sensorless control method for synchronous machines with a parallel model are investigated, in consideration of additional online parameter adaptation. Using error compensated reference voltage inputs the parallel model's parameters are adapted to the synchronous machine by comparing the model current output with the measured current values. Thus the adaptation to the reference transfer behavior of the system is accomplished using an implicit method. After describing the difference between the reference and the adaptive model mathematically, the system stability is analysed using Popov's hyperstability condition. An additional stability criterion could be derived from the full machine model. The extension of the sensorless control was achieved through a simultaneous adaptation algorithm accounting for coupling effects caused by the influence of misalignment of the field-coordinates.

## I. Introduction

The origin of the Model Reference Adaptive System (MRAS) scheme is related to a series of research projects for adaptive control systems. It was in this context that Whitaker suggested the name MRAS [1]. Following significant improvements in various types of structures and their stability, the term MRAS has become popular in the field of sensorless drives [2], [3], [4]. In this context, the term MRAS indicates that

elements of the proposed method are derived from the foundational ideas introduced by leading researchers in this field [1]. Referring to these developments, MRAS-based methods have proven to be particularly advantageous for sensorless control applications. They offer several benefits, including computational efficiency, adaptability to parameter variations, and good dynamic response. In their basic form, they inherently determine position through an integrator, eliminating the need for additional signal filtering. These characteristics make them well-suited for real-time implementation on microcontrollers.

Axial flux permanent magnet synchronous machines (AFPMSM), which have advantages in comparison to radial flux machines, particularly at low-speed, high-torque applications, are becoming popular in the field of electrical drive systems. This makes them especially attractive for electric mobility, mobile machine tools and specialized industrial applications requiring high power densities. However, current disadvantages lie in the complexity of maintaining an optimal air gap and the need for high precision at new manufacturing processes, in order to prevent parameter variations and performance fluctuations. But recent advances in manufacturing technologies are beginning to address these issues [5], [6].

While MRAS-based methods perform well at medium and high stator frequencies, model inaccuracies and poor signal-to-noise ratio can significantly limit their performance, especially under varying operational conditions.

There are many alternative proposed methods that enable sensorless control in medium and high-speed ranges, but no standardized approach exists
for low-speed operation down to zero speed without signal injection. References [7] and [8] still provide a valuable overview of the current state of the art, despite having been published at the beginning of the decade.

The objective of this work is to efficiently describe the stability of the MRAS-based method. The analyzed mathematical tools should provide a basis for the justified extension of the applicable speed range without additional signal processing.

## II. MRAS-based method

The adaptation mechanism of the MRAS-based method is fed by the difference between state variables of the reference and the adjustable parallel model. Both reference system and model are supplied with identical voltage input values using error compensated reference voltages. In the investigated MRAS-based method, the controlled machine itself serves as reference system and the measured stator currents are used to adjust the parallel model as illustrated in Fig. 1. In addition to the published approach in [9] a simultaneous parameter adaptation could be employed.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Scheme of MRAS-based sensorless control for AFPMSM with parameter adaptation</figcaption></figure>

### a. Reference model of AFPMSM

Due to the non-linear characteristics of the AFPMSM, which are not fully described by the fundamental wave model in rotor dq-coordinates, the modal transformation is used to expose neglected effects such as winding asymmetries or magnetic saturation [10]. Fig. 2 presents the three-phase equivalent circuit of an AFPMSM consisting of the inductances L<sub>a,b,c</sub>, the mutual inductances M<sub>ab,bc,ac</sub> between the phases U, V, and

W, the phase resistances R<sub>S_a,b,c</sub>, the induced voltages u<sub>P_a,b,c</sub> due to the permanent magnet flux linkages Ψ<sub>PM_a,b,c</sub>, the source voltages u<sub>S_a,b,c</sub>, the neutral point voltage difference u<sub>N</sub> and the stator currents i<sub>S_a,b,c</sub>.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Three-phase equivalent circuit of an AFPMSM [10]</figcaption></figure>

$$ \underline{R}_{S\_abc} = \begin{bmatrix} R_a & 0 & 0 \\ 0 & R_b & 0 \\ 0 & 0 & R_c \end{bmatrix}; \quad \underline{L}_{S\_abc} = \begin{bmatrix} L_a & M_{ab} & M_{ac} \\ M_{ab} & L_b & M_{bc} \\ M_{ac} & M_{bc} & L_c \end{bmatrix} \quad (1) $$

$$ \underline{\psi}_{PM\_abc} = \begin{bmatrix} \psi_{PMa} \\ \psi_{PMB} \\ \psi_{PMC} \end{bmatrix}; \quad \underline{u}_{S\_abc} = \begin{bmatrix} u_{Sa} \\ u_{Sb} \\ u_{Sc} \end{bmatrix}; \quad \underline{i}_{S\_abc} = \begin{bmatrix} i_{Sa} \\ i_{Sb} \\ i_{Sc} \end{bmatrix} \quad (2) $$

The decoupling of the original variables from Fig. 2 under transient conditions can be carried out using modal components, based on the assumption in Eq. (3) and the power-variant transformation matrix in Eq. (4), which is defined for rotor-fixed rotating dq-coordinates, taking into account the rotor flux angle ε [10].

$$ \underline{i}_{S\_abc} = \underline{T}_{dq} \cdot \underline{i}_{M}; \quad \underline{R}_{M} = \underline{T}_{dq}^{-1} \cdot \underline{R}_{S\_abc} \cdot \underline{T}_{dq} \quad (3.a) $$

$$ \underline{u}_{S\_abc} - \underline{u}_{N} = \underline{T}_{dq} \cdot \underline{u}_{M}; \quad \underline{L}_{M} = \underline{T}_{dq}^{-1} \cdot \underline{L}_{S\_abc} \cdot \underline{T}_{dq} \quad (3.b) $$

$$ \underline{\psi}_{PM\_abc} = \underline{T}_{dq} \cdot \underline{\psi}_{MPM} \qquad \frac{d\varepsilon}{dt} = \omega_e \quad (3.c) $$

$$ \underline{T}_{dq} = \begin{pmatrix} \cos(\varepsilon) & -\sin(\varepsilon) & 1 \\ \cos(\varepsilon - \frac{2}{3}\pi) & -\sin(\varepsilon - \frac{2}{3}\pi) & 1 \\ \cos(\varepsilon + \frac{2}{3}\pi) & -\sin(\varepsilon + \frac{2}{3}\pi) & 1 \end{pmatrix} \quad (4) $$

$$ \underline{L}_{M} \cdot \frac{d\underline{i}_{M}}{dt} = \underline{u}_{M} - \underline{R}_{M} \cdot \underline{i}_{M} \\ - \left[ \underline{T}_{dq}^{-1} \cdot \frac{d\underline{L}_{M}}{dt} \cdot \underline{T}_{dq} + \left( \underline{L}_{M} \cdot \underline{T}_{dq}^{-1} \cdot \frac{d\underline{T}_{dq}}{dt} \right) \right] \cdot \underline{i}_{M} \quad (5) \\ - \underline{T}_{dq}^{-1} \cdot \frac{d\underline{T}_{dq}}{dt} \cdot \underline{\psi}_{MPM} - \frac{d\underline{\psi}_{MPM}}{dt} $$

In Eq. (5), coupled with the saturation effects and unbalanced supply voltage, an asymmetry in the parameters is accounted for [11]. Consequently, the respective modal parameter matrices, e.g. R<sub>M</sub>,
are influenced by oscillation components that depend on both the fundamental stator frequency for the zero-sequence component and the double stator frequency for the d- and q-axis components, in the presence of imbalance. Because the motor is connected in y-configuration and a neutral conductor ZN is usually omitted, only the dq-components are considered. The reference model can be described in the state-space Eq. (7) form using the stator current equation Eq. (6), oriented to the rotor flux.

$$ 
\begin{aligned}
\dot{x}_{MRM} = \frac{di_M}{dt} = \underline{L}_M^{-1} \cdot \{u_M - \underline{R}_M \cdot \dot{i}_M \\
- \left[ \frac{d\underline{L}_M}{dt} + \left( \underline{L}_M \cdot \frac{d\epsilon}{dt} \cdot \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \right) \right] \cdot \dot{i}_M \\
- \frac{d\epsilon}{dt} \cdot \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \cdot \underline{\psi}_{MPM} - \frac{d\underline{\psi}_{MPM}}{dt} \}
\end{aligned}
\quad (6) $$

$$ \dot{x}_{MRM} = \underline{A}_{MRM} \cdot \underline{x}_{MRM} + \underline{B}_{MRM} \cdot \underline{u}_{MRM} + \underline{Z}_{MRM} + \underline{Z}_{M\Delta\epsilon} \quad (7) $$

$$ \underline{y}_{MRM} = \underline{x}_{MRM} \quad (8) $$

The disturbance vector $\underline{Z}_{MRM}$ includes all deviations from the fundamental wave model, such as phase asymmetries and saturation effects. Additionally, the disturbance vector $\underline{Z}_{M\Delta\epsilon}$ accounts for errors due to misalignment or incorrect assumptions about the rotor flux position. The error angle $\Delta\epsilon$ represents the difference between the actual rotor flux angle and the assumed angle $\hat{\epsilon}$ used for the modal transformation. With the introduction of the disturbance vectors, further calculations can now be performed using the average stator phase resistance $R_S$, the average stator phase inductance $L_S$, and the average phase permanent flux linkage $\Psi_{PM}$.

$$ \underline{u}_{MRM} = \begin{bmatrix} u_{Md} \\ u_{Mq} - \omega_e \cdot \Psi_{PM} \end{bmatrix}, \qquad \underline{x}_{MRM} = \begin{bmatrix} i_{Md} \\ i_{Mq} \end{bmatrix} \quad (9.a) $$

$$ \underline{B}_{MRM} = \begin{bmatrix} \frac{1}{L_S} & 0 \\ 0 & \frac{1}{L_S} \end{bmatrix}; \qquad \underline{A}_{MRM} = \begin{bmatrix} -\frac{R_S}{L_S} & \omega_e \\ -\omega_e & -\frac{R_S}{L_S} \end{bmatrix} \quad (9.b) $$

$$ \underline{Z}_{M\Delta\epsilon} = \begin{bmatrix} -\frac{R_S}{L_S} & \omega_e \\ -\omega_e & -\frac{R_S}{L_S} \end{bmatrix} \cdot \begin{bmatrix} i_{\Delta d} \\ i_{\Delta q} \end{bmatrix} - \begin{bmatrix} \frac{d\Delta i_d}{dt} \\ \frac{d\Delta i_q}{dt} \end{bmatrix} + \begin{bmatrix} \frac{1}{L_S} & 0 \\ 0 & \frac{1}{L_S} \end{bmatrix} \cdot \begin{bmatrix} u_{\Delta d} \\ u_{\Delta q} \end{bmatrix} + \begin{bmatrix} \frac{\Psi_{PM}}{L_S} \cdot \sin(\Delta\epsilon) \\ \frac{\Psi_{PM}}{L_S} (1 - \cos(\Delta\epsilon)) \end{bmatrix} \cdot \omega_e \quad (9.c) $$

$$ \begin{bmatrix} i_{\Delta d} \\ i_{\Delta q} \end{bmatrix} = \begin{bmatrix} (\cos(\Delta\epsilon) & -\sin(\Delta\epsilon) \\ \sin(\Delta\epsilon) & \cos(\Delta\epsilon) \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} i_{Md} \\ i_{Mq} \end{bmatrix} \quad (9.d) $$

$$ \begin{bmatrix} u_{\Delta d} \\ u_{\Delta q} \end{bmatrix} = \begin{bmatrix} (\cos(\Delta\epsilon) & -\sin(\Delta\epsilon) \\ \sin(\Delta\epsilon) & \cos(\Delta\epsilon) \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} u_{Md} \\ u_{Mq} \end{bmatrix} \quad (9.e) $$

$$ \Delta\epsilon = \epsilon - \hat{\epsilon} \qquad R_S = \frac{R_a + R_b + R_c}{3} \quad (9.f) $$

$$ L_S = \frac{L_a + L_b + L_c - M_{ab} - M_{bc} - M_{ca}}{3} \quad (9.g) $$

$$ \Psi_{PM} = \frac{\Psi_{PMb} + \Psi_{PMB} + \Psi_{PMC}}{3} \quad (9.h) $$

## b. Adjustable parallel model

The adjustable model, see Fig. 1, is designed as the fundamental wave model in state-space form, which allows for efficient implementation on computing architectures.

$$ \dot{x}_{AM} = \underline{A}_{AM} \cdot \underline{x}_{AM} + \underline{B}_{AM} \cdot \underline{u}_{AM} \quad (10) $$

$$ \underline{y}_{AM} = \underline{x}_{AM} \quad (11) $$

The circumflex indicates that the parameters can be adapted during operation as needed, since this adaptation was conducted based on stability criteria. These parameters, such as $\hat{R}_S$, retain the same physical meaning as in the reference model, while the adjustable parallel model assumes a perfectly symmetrical machine.

$$ \underline{u}_{AM} = \begin{bmatrix} u_{Md} \\ u_{Mq} - \hat{\omega}_e \cdot \hat{\Psi}_{PM} \end{bmatrix}, \qquad \underline{x}_{AM} = \begin{bmatrix} \hat{i}_d \\ \hat{i}_q \end{bmatrix} \quad (12.a) $$

$$ \underline{B}_{AM} = \begin{bmatrix} \frac{1}{\hat{L}_S} & 0 \\ 0 & \frac{1}{\hat{L}_S} \end{bmatrix}; \qquad \underline{A}_{AM} = \begin{bmatrix} -\frac{\hat{R}_S}{\hat{L}_S} & \hat{\omega}_e \\ -\hat{\omega}_e & -\frac{\hat{R}_S}{\hat{L}_S} \end{bmatrix} \quad (12.b) $$

## c. Approach of error model theory

The adaptation in the MRAS-based method is achieved through the feedback of the difference between the output of the reference and adjustable model, as defined in Eq. (13). To ensure system
stability, this feedback is typically processed through a non-linear feedback matrix C<sub>EM</sub> designed based on Popov's hyperstability criterion, in conjunction with a proportional-integral (PI) controller [12].

$$ \dot{\underline{x}}_{EM} = \dot{\underline{\delta}} = \dot{\underline{x}}_{MRM} - \dot{\underline{x}}_{AM} \quad (13) $$

$$ \underline{y}_{EM} = \underline{C}_{EM} \cdot \underline{x}_{EM} \quad (14) $$

An important property of hyperstable systems is that if a hyperstable system is connected in feedback with another hyperstable system, the entire system remains hyperstable. Due to this property, the error model can be split into a forward and a feedback path, which can be analysed independently [13]. Furthermore, with a suitable system setup, the Kalman-Yakubovich-Popov (KYP) lemma can be applied more easily to one part of the system. This means that a part is hyperstable or asymptotically hyperstable whenever it can be shown that its transfer matrix G(s) is positive real or strictly positive real [2], [14].

$$ \underline{A}_{MRM} - \underline{\hat{A}}_{AM} = \underline{\Delta A} \rightarrow \underline{\hat{A}}_{AM} = \underline{A}_{MRM} - \underline{\Delta A} \quad (15) $$

$$ \underline{B}_{MRM} - \underline{\hat{B}}_{AM} = \underline{\Delta B} \rightarrow \underline{B}_{MRM} = \underline{\hat{B}}_{AM} + \underline{\Delta B} \quad (16) $$

In consideration of Eq. (13)-(16) an error model is obtained, which can be subdivided into a part depending on real system parameter, and a part that includes the adaptive components of the adjustable model, and a disturbance part.

$$ \begin{aligned} \dot{\underline{\delta}} = & \underline{A}_{MRM} \cdot \underline{\delta} \\ & + \underline{\hat{B}}_{AM} \cdot \underline{\Delta u} + \underline{\Delta A} \cdot \underline{x}_{AM} + \underline{\Delta B} \cdot (\underline{\Delta u} + \underline{\hat{u}}_{AM}) \\ & + \underline{z}_{MRM} + \underline{z}_{M\Delta\epsilon} \end{aligned} \quad (17) $$

Assuming that no simultaneous parameter deviations occur initially, a further simplification can be made. Thus, the electrical part of the error model could finally be described by Eq. (18)-(21). This result forms the basis for determining the output matrix applied for stability theory.

$$ \underline{\delta} = \begin{bmatrix} \delta_d \\ \delta_q \end{bmatrix} = \begin{bmatrix} i_{Md} - \hat{i}_d \\ i_{Mq} - \hat{i}_q \end{bmatrix} \quad (18) $$

$$ \dot{\underline{\delta}} = \underline{A}_{MRM} \cdot \underline{\delta} + \underline{B}_{EM} \cdot \underline{\Delta u}_{EM} + \underline{z}_{MRM} + \underline{z}_{M\Delta\epsilon} \quad (19) $$

$$ \underline{y}_{EM} = \underline{C}_{EM} \cdot \underline{\delta} \quad (20) $$

### d. Stability of feedback error model

It is well known that the transfer matrix G<sub>HEM</sub> of the forward part of the error model is strictly positive real [14].

$$ \begin{aligned} \underline{G}_{HMRM}(s) &= (s\underline{I} - \underline{A}_{MRM})^{-1} \\ &= \begin{bmatrix} s + \frac{R_s}{L_s} & \omega_e \\ -\omega_e & s + \frac{R_s}{L_s} \end{bmatrix} \cdot \sigma_1 \\ \sigma_1 &= \frac{1}{(s + \frac{R_s}{L_s})(s + \frac{R_s}{L_s}) + \hat{\omega}_e^2} \end{aligned} \quad (21) $$

Overall, good results have been achieved with this approach, through the specific construction of a linear forward part and a non-linear feedback path. Furthermore, it must still be verified whether the stability of the non-linear feedback path arising in this context is ensured. This is typically done using the Popov integral inequality, which is generalized as concept of dissipative systems [15]. The disadvantage is the interpretation of the complex result [3], [9], [12].

$$ \int_{0}^{t_0} (\underline{B}_{EM}(\tau) \cdot \underline{\Delta u}_{EM}(\tau))^T \cdot \underline{\delta}(\tau) \, d\tau \geq -\gamma_0^2 \quad (22) $$

The generally presented Popov stability condition can be further simplified using indefinite integrals and the relation B<sub>EM</sub><sup>T</sup>=C<sub>EM</sub>, allowing the stability boundary for speed adaptation of an AFPMSM according to Popov inequality to be expressed as a function of a PI controller, where K<sub>P</sub> represents the proportional gain and T<sub>N</sub> the integral time constant.

$$ \begin{aligned} & \frac{-1}{T_N} \cdot \left[ \int_0^t (B_{EM,11} \cdot \delta_1 + B_{EM,21} \cdot \delta_2) \cdot d\tau - T_N \cdot \omega_e \right]^2 \\ & -2 \cdot K_P \cdot \int_0^t (B_{EM,11} \cdot \delta_1 + B_{EM,21} \cdot \delta_2)^2 \cdot d\tau \\ & \leq 2 \cdot \epsilon_0^2 - T_N \cdot \omega_e^2 \end{aligned} \quad (23) $$

Alternatively, the hyperstability of the error model in Eq. (19)-(20) is proven if real matrices L, V, and P satisfy the system of Eq. (24)-(25) where L and V are arbitrary and exist, but P must be positive definite. If L is also regular, the dynamic system is asymptotically hyperstable [2] [14].

$$ \underline{A}_{EM}^T \cdot \underline{P} + \underline{P} \cdot \underline{A}_{EM} = -\underline{L} \cdot \underline{L}^T \quad (24) $$
$$ \underline{L} \cdot \underline{V} = \underline{C}_{EM}^T - \underline{P} \cdot \underline{B}_{EM} \qquad (25) $$

One possible solution to the KYP lemma is presented in Eq. (26).

$$ \underline{P} = \begin{bmatrix} \frac{L_S}{R_S} & 0 \\ 0 & \frac{L_S}{R_S} \end{bmatrix}, \underline{V} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \qquad (26.a) $$

$$ \underline{L} \cdot \underline{L}^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad (26.b) $$

Assuming that Eq. (28) is satisfied, the input matrix of the error model can be calculated from Eq. (17).

$$ \underline{B}_{EM} = \begin{bmatrix} \hat{i}_q & \frac{-\hat{i}_d}{\hat{L}_s} & 0 & \hat{R}_S \cdot \hat{i}_d - u_{dM} \\ \frac{-\hat{\Psi}_{PM}}{\hat{L}_s} - \hat{i}_d & \frac{-\hat{i}_q}{\hat{L}_s} & \frac{-\hat{\omega}_e}{\hat{L}_s} & \hat{R}_S \cdot \hat{i}_q - u_{qM} + \hat{\omega}_e \cdot \hat{\Psi}_{PM} \end{bmatrix} \qquad (27) $$

$$ \Delta \underline{u}_{EM} = \begin{bmatrix} \frac{\Delta \omega_e}{\Delta R_S} \\ \frac{\Delta \Psi_{PM}}{\Delta L_S} \\ \frac{\Delta R_S}{\Delta L_S + \hat{L}_S} \end{bmatrix} + \underline{G}_{EM} \cdot \underline{Z}_{EM} \cdot (\underline{B}_{EM})^{-1} \qquad (28) $$

$$ \begin{aligned} |\Delta\omega_e| &> 0 \approx \Delta\Psi_{PM} \approx \Delta L_S \\ |\Delta R_S| &> 0 \approx \Delta\Psi_{PM} \approx \Delta L_S \\ |\Delta\Psi_{PM}| &> 0 \approx \Delta\omega_e \approx \Delta L_S \approx \Delta R_S \\ L_S &> |\Delta L_S| > 0 \approx \Delta\omega_e \approx \Delta R_S \approx \Delta\Psi_{PM} \end{aligned} $$

With respect to Eq. (25)-(27) the output matrix $\underline{C}_{EM}$ can be determined in such a way that the transfer matrix of the error model becomes asymptotically hyperstable.

$$ \underline{C}_{EM} = \underline{B}_{EM}^T \cdot \underline{P}^T \qquad (29) $$

### III. Novel feedback error model

The approach presented in Fig. 3 below offers advantages in the synthesis of the feedback controller matrix $\underline{K}_{EM}$ in relation to the traditional approach, which relies on a linear forward path and a nonlinear feedback path. These advantages arise from the fact that only a strictly positive real or positive real transfer matrix is required. The approach is not bound to a specific controller type. Furthermore, the system shift creates a structure resembling the traditional multivariable control configuration, enabling the controller synthesis to be performed in a similar manner.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Feedback error model for parameter adaptation in MRAS-based method considering disturbance effects</figcaption></figure>

In the case of parameter mismatches the error model is fed with an input $\Delta \underline{U}_{EM}$. Therefore the dynamic error model is stimulated. Due to the controller feedback path the system reacts ensuring that the input vector $\Delta \underline{U}_{EM}$ is driven to zero. Additionally, the previously described disturbances are taken into account, revealing that if a disturbance becomes sufficiently large and exhibits an angle between 90° and 270° relative to the model reference matrix, coupling effects may occur, potentially rendering the system unstable. For improved clarity of the method and a better correlation between Fig. 3 and Fig. 1, the parameter vectors $\hat{\underline{P}}_{\omega_e}$, $\hat{\underline{P}}_P$ and $\underline{P}_{\omega_e}$ are introduced:

$$ \underline{P}_{\omega_e} = \begin{bmatrix} \hat{\omega}_e \\ \hat{R}_S \\ \hat{\Psi}_{PM} \\ \hat{L}_S \end{bmatrix}, \qquad \underline{P}_P = \begin{bmatrix} \hat{R}_S \\ \hat{\Psi}_{PM} \\ \hat{L}_S \end{bmatrix}, \qquad \underline{P}_{\omega_e} = \begin{bmatrix} \omega_e \\ R_S \\ \Psi_{PM} \\ L_S \end{bmatrix} \qquad (30) $$

This result highlights an advantage of the presented approach, as it becomes evident from the structure and explains an instability effect associated with the investigated MRAS-based method under conditions of parameter mismatch [16].

#### a. Extended stability criterion

Although the error model is hyperstable, the stability of the field oriented controlled AFPMSM using the MRAS-based method can only be guaranteed within a certain range. This limitation is defined by the maximum specified load torque $M_w$, and the machine-specific pull-out angle $\theta_p$, both of which must be managed despite the presence of an error angle $\Delta\epsilon$.
$$
\begin{align}
|M_W| < |M_{e\_Max}| \cdot \sin(\vartheta_p - \Delta\varepsilon) \nonumber \\
\vartheta_{p,mot} &= -\frac{\pi}{2}; R_S = 0; L_d = L_q \tag{31} \\
\vartheta_{p,gen} &= \frac{\pi}{2}; R_S = 0; L_d = L_q \nonumber
\end{align}
$$

The error angle is obtained by integrating the
difference between the actual speed and the speed
estimated by the MRAS-based method. In this
method, the reference speed is derived from the
electrical torque M<sub>e</sub> in field coordinates, with θ
representing the mass moment of inertia.

$$
\Omega = \frac{1}{\theta} \cdot \int \left( \frac{3}{2} \cdot p \cdot \Psi_{PM} \cdot i_q - M_W \right) dt \quad (32)
$$

By employing the MRAS-based method to
determine the rotational speed, all factors
influencing the stability of the sensorless control
strategy can be systematically identified.

$$
\hat{\Omega}(t) = K_{EM,11}(t) \cdot \frac{1}{p} \frac{L_s}{R_s} \left( \frac{\hat{\Psi}_{PM}}{\hat{L}_s} \delta_q + \hat{i}_d \delta_q - \hat{i}_q \delta_d \right) \quad (33)
$$

$$
\Delta\epsilon = p \cdot \int (\Omega(t) - \hat{\Omega}(t)) \cdot dt \quad (34)
$$

$$
\Delta\epsilon < \vartheta_p - \arcsin\left(\frac{|M_W|}{|M_{e\_Max}|}\right) \quad (35)
$$

The combination of Eq. (32) to Eq. (34) introduces additional requirements for the controller matrix K<sub>EM</sub> to ensure compliance with Eq. (35). This novel stability approach, introduced in combination with the MRAS-based method, enables an exact determination of the overall system boundary. While the instability of synchronous machines at certain pole angles is well-known, this aspect is not typically included in the stability analysis of the MRAS-based method, offering an alternative for controller synthesis compared to the Popov integral inequality in Eq. (22).

IV. Simulation and measurements

This paper presents an analysis of the adaptation
of a parallel model for an AFPMSM using the
discussed MRAS method in MATLAB/Simulink.
Additionally, initial measurements were
conducted on the test bench to adapt the stator
resistance during sensorless operation.

a. Simulation results

Simulation results demonstrate the performance
of the adaptation depending on the speed
trajectory. For the controller matrix K<sub>EM</sub>, a PI
approach with constant values was selected.

The simultaneous adaptation of speed and stator
resistance performs well up to the medium speed
range, whereas undesirable oscillations may
occur, particularly at lower speeds. Fig. 4
illustrates the simultaneous adaptation of stator
resistance during sensorless operation at nominal
load of 0.7 Nm and the resulting error angle.
Additionally, during generator operation, a
coupling effect between the influence of the error
angle and the adaptation of the stator resistance
was observed. This demonstrates that the
influence of the disturbance can become
significant at certain operating points, potentially
leading to instabilities.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Adaptation of stator resistance during
sensorless operation at M<sub>w</sub>=0,7Nm</figcaption></figure>

a) Electrical angular frequency
estimated by MRAS compared to the
measured value

b) Flux-orientation error

c) Stator resistance estimated by MRAS compared to the machine value

Countermeasures against coupling effects can be
derived from Fig. 3. Among them, one approach
extends the adaptation of the controller for $\hat{R}_S$ at specific operating points by applying a 180° phase shift to the relevant part of the output vector, see Eq. (20). Fig. 5 demonstrates that this strategy enables effective adaptation even during sensorless generator operation.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Adaptation of the stator resistance at $M_W=0,7$Nm, including a phase shift of the input from the $\hat{R}_S$ controller during generator operation</figcaption></figure>

a) Electrical angular frequency
estimated by MRAS compared to the
measured values

b) Flux-orientation error

c) Stator resistance estimated by MRAS
compared to the machine values

Based on the hypothesis that all parameters remain identical except for one, Fig. 6 and Fig. 7 illustrate the adaptation of the inductance and magnetic flux. Although inductance adaptation during sensorless operation poses a challenge according to Eq. (28), it can succeed under certain conditions, Fig. 6. Due to nonlinear, operating-point-dependent adaptation, the error is not corrected with constant dynamics, and from 2.5 s onward, adaptation in Fig. 6 slows significantly. Even with a position sensor, e.g., for parameter identification, operating-point-dependent dynamics persist, as shown in the estimation of the permanent magnet flux linkage in Fig. 7.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Adaptation of stator inductance during sensorless operation at $M_W=0,7$Nm</figcaption></figure>

a) Electrical angular frequency
estimated by MRAS compared to the
measured values

b) Flux-orientation and parameter error
of the stator inductance $L_S$

c) Stator inductance estimated by MRAS
compared to the machine values

<figure><img src="image_8.png" /><figcaption>Fig. 7: Adaptation of the magnetic flux linkage using a position sensor at $M_W=0,7$Nm</figcaption></figure>

a) Measured values of electrical angular frequency measured

b) Permanent magnet flux linkage
estimated by MRAS compared to the
machine values
The oscillations observed in the simulation results are caused by the interaction between parameter variations, simultaneous adaptations, and the disturbance vector $\Sigma_{MA\epsilon}$. They occur primarily at the transition edges of the reference values. However, they can persist in the presence of both parameter deviations and an unfavorably chosen controller gains. In such cases, as depicted in Fig. 7, sensor-based control can effectively eliminate disturbances, resulting in reduced oscillations.

## b. Measurement results

The experimental setup shown in Fig. 8 was used to validate the proposed method by measuring the performance of the AFPMSM from Heinzmann, type PMS066F, rated at 24 VDC / 3000 rpm / 0.35 kW, during operation. The identified machine parameters are: $L_s=0.172$ mH, $R_{s25°}=0.1716$ Ω, $\Psi_{PM}=12,5$ mVs, five pole pairs and $\theta=3,16 \cdot 10^{-5}$ kg*m².

<figure><img src="image_9.png" /><figcaption>Fig. 8: Structure of the test bench setup. 1: PMS066F; 2: shaft coupling; 3: torque sensor; 4: load machine; 5: position and temperature sensor; 6: Plexim RT Box 2; 7: Texas Instruments DRV8305; 8: DC source; 9: power analyzer; 10: Inverter; 11: control unit</figcaption></figure>

Measurement results at the half of nominal load torque show the effects of the presented MRAS-based sensorless control method with simultaneous resistor adaptation under designed operating temperature variations.

<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9: Adaptation of stator resistance during sensorless operation at $M_w=M_N/2$</figcaption></figure>

a) Electrical angular frequency estimated by MRAS compared to the measured value

b) Flux-orientation error

c) Stator resistance estimated by MRAS compared to the temperature-compensated stator resistance

d) Measured stator winding temperature using a mounted KTY temperature sensor

Fig. 9 shows that under real conditions, more than just adapted parameter uncertainty occurs, resulting in a small error angle and oscillations during sensorless operation. The adapted speed could be utilized directly for the speed control difference and the resulting angle for the transformation in model field coordinates. Despite temperature-induced changes in the stator resistance, the error angle remains constant due to the additionally implemented resistance adaptation, even in the presence of unknown model deviations. The resistance value calculated through MRAS shows a deviation of
approximately 2% compared to the temperature dependent resistance determined using the temperature coefficient $α_{Cu}=0.0039 \text{ K}^{-1}$.

V. Conclusion

An MRAS-based sensorless control strategy designed for medium and high stator frequencies was investigated for AFPMSM, focused on additional parameter adaptation to reduce effects due to parameter variations.

Initially, the sensorless control method was introduced. The effects of multiple parameter variations are described by a mathematical error model. An alternative approach based on hyperstability theory and the KYP lemma is proposed to design the controller structure of the adaptation mechanism independently from the Popov integral inequality. Additionally, the influence of an orientation error angle on the field oriented variables was considered, which justifies the adaptation of the parameter adaptation mechanisms under stability considerations.

The simulation results demonstrate that the adaptation performance of the considered parameters depends on the operating point and the error angle of flux-orientation. Furthermore, the measurement results show that the additional resistance adaptation can effectively mitigate the error angle caused by temperature-dependent resistance under real operating conditions of an MRAS-based, sensorless, field oriented AFPMSM control.
## References

[1] K. Aström, 'History of Adaptive Control', in *Encyclopedia of Systems and Control*, J. Baillieul and T. Samad, Eds., London: Springer, 2014, pp. 1–9. doi: 10.1007/978-1-4471-5102-9_120-1.

[2] K. Liu, Q. Zhang, Z.-Q. Zhu, J. Zhang, A.-W. Shen, and P. Stewart, 'Comparison of two novel MRAS based strategies for identifying parameters in permanent magnet synchronous motors', *Int. J. Autom. Comput.*, vol. 7, no. 4, pp. 516–524, Nov. 2010, doi: 10.1007/s11633-010-0535-3.

[3] Y. Zhang, Z. Yin, X. Sun, and Y. Zhong, 'Online identification methods of parameters for permanent magnet synchronous motors based on cascade MRAS', in *2015 9th International Conference on Power Electronics and ECCE Asia (ICPE-ECCE Asia)*, Jun. 2015, pp. 345–350. doi: 10.1109/ICPE.2015.7167808.

[4] Z. D. An et al., 'An Improved Model Predictive Current Control of Permanent Magnet Machine with Integrated MRAS-based Online Parameter Identification', in *2023 26th International Conference on Electrical Machines and Systems (ICEMS)*, Nov. 2023, pp. 5132–5136. doi: 10.1109/ICEMS59686.2023.10344331.

[5] F. Nishanth, J. Van Verdeghem, and E. L. Severson, 'A Review of Axial Flux Permanent Magnet Machine Technology', *IEEE Trans. Ind. Appl.*, vol. 59, no. 4, pp. 3920–3933, Jul. 2023, doi: 10.1109/TIA.2023.3258933.

[6] A. Parviainen, *Design of axial-flux permanent-magnet low-speed machines and performance comparison between radial-flux and axial-flux machines*. in Acta Universitatis Lappeenrantaensis / Lappeenrannan Teknillinen Yliopisto, no. 208. Lappeenranta: Lappeenrannan Teknillinen Yliopisto, 2005.

[7] G. Wang, M. Valla, and J. Solsona, 'Position Sensorless Permanent Magnet Synchronous Machine Drives—A Review', *IEEE Trans. Ind. Electron.*, vol. 67, no. 7, pp. 5830–5842, Jul. 2020, doi: 10.1109/TIE.2019.2955409.

[8] D. Xiao, S. R. Filho, G. Fang, J. Ye, and A. Emadi, 'Position-Sensorless Control of Switched Reluctance Motor Drives – A Review', *IEEE Trans. Transp. Electrification*, pp. 1–1, 2021, doi: 10.1109/TTE.2021.3110867.

[9] M. Brüns, C. Rudolph, and T. Müller, 'Parameter sensitivity of a MRAS-based sensorless control for AFPMSM considering speed accuracy and

dynamic response at multiple parameter variations', in *2022 24th European Conference on Power Electronics and Applications (EPE'22 ECCE Europe)*, Sep. 2022, pp. 1–10.

[10] *Electric power engineering - Modal components in three-phase a.c. systems - Quantities and transformations (IEC 62428:2008); German version EN 62428:2008*, DIN standard DIN EN 62428, Feb. 01, 2009.

[11] T. Werner, *Geberlose Rotorlagebestimmung in elektrischen Maschinen*. Wiesbaden: Springer Fachmedien, 2018. doi: 10.1007/978-3-658-22271-0.

[12] A. Khlaief, M. Boussak, and A. Châari, 'A MRAS-based stator resistance and speed estimation for sensorless vector controlled IPMSM drive', *Electr. Power Syst. Res.*, vol. 108, pp. 1–15, Mar. 2014, doi: 10.1016/j.epsr.2013.09.018.

[13] H. Unbehauen, *Regelungstechnik III*. Wiesbaden: Vieweg+Teubner Verlag, 2000. doi: 10.1007/978-3-322-94391-0.

[14] O. Föllinger, 'Nichtlineare Regelungen 2: Harmonische Balance, Popow- und Kreiskriterium, Hyperstabilität, Synthese im Zustandsraum: mit 18 Übungsaufgaben mit Lösungen', in *Nichtlineare Regelungen 2*, De Gruyter Oldenbourg, 2014. doi: 10.1524/9783110406146.

[15] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland, *Dissipative Systems Analysis and Control: Theory and Applications*. Cham, SWITZERLAND: Springer International Publishing AG, 2019. Accessed: Jan. 15, 2025. [Online]. Available: http://ebookcentral.proquest.com/lib/hawhamburg-ebooks/detail.action?docID=5811736

[16] Y. Ni and D. Shao, 'Research of Improved MRAS Based Sensorless Control of Permanent Magnet Synchronous Motor Considering Parameter Sensitivity', in *2021 IEEE 4th Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC)*, Jun. 2021, pp. 633–638. doi: 10.1109/IMCEC51613.2021.9482131.