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# Sensorless IPMSM Drive Enhanced with Parameter Identification Using Open-Loop Current Prediction Error and Gradients

Aravinda Perera, Roy Nilsen

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## To cite this version:

Aravinda Perera, Roy Nilsen. Sensorless IPMSM Drive Enhanced with Parameter Identification Using Open-Loop Current Prediction Error and Gradients. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0135. hal-05099804

HAL Id: hal-05099804

https://utc.hal.science/hal-05099804v1

Submitted on 5 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Sensorless IPMSM Drive Enhanced with Parameter Identification Using Open-Loop Current Prediction Error and Gradients

Aravinda Perera, Roy Nilsen

Institute of Electric Energy

Norwegian University of Science and Technology

Trondheim, Norway

Tel.: +47 73 59 42 10

E-Mail: aravinda.perera@ntnu.no, roy.nilsen@ntnu.no

URL: https://www.ntnu.no/iel/institutt-for-elektrisk-energi

**Index Terms**—Hessian, identification, PMSM, recursive prediction, sensorless control, variable speed drive.

**Abstract**—This article presents a position-sensorless Interior Permanent Magnet Synchronous Machine (IPMSM) drive scheme that incorporates an open-loop stator current predictor to identify electrical parameters and improve observer performance. Unlike classical observers, the proposed open-loop, full-order model is inherently sensitive to discrepancies between the physical and model parameters. Capitalizing on this sensitivity through current prediction error gradients (PEGs), the parameter estimator scheme identifies the temperature sensitive parameters, i.e., the permanent magnet flux linkage $\Psi_m$ and stator resistance $R_s$. An offline experimental method, utilizing the same prediction model, is proposed for mapping the inductance-current relationship, which is subsequently implemented in the processor using a look-up table (LUT). Rotor position and speed are estimated via an Active Flux Observer, with real-time updates to all model parameters. Furthermore, a novel approach for tuning the estimation gain matrix using customized Hessian functions is presented, enabling improved simultaneous identification of $\Psi_m$ and $R_s$. Experimental validation is conducted on a 3 kW IPMSM drive, with control and estimation routines implemented on a Zynq System-on-Chip (SoC)-based industrial embedded control platform. Results demonstrate that the proposed schemes enhance observer stability, precision, and robustness to parameter variations compared to conventional observers without parameter adaptation.

## I. INTRODUCTION

POSITION sensorless Interior Permanent Magnet Synchronous Machine (IPMSM) drives are becoming state-of-the-art in high-performance and mission-critical applications due to several attractive features, including superior efficiency, high power density, and enhanced

reliability [1], [2]. Among the methods for sensorless control [3], [4], fundamental excitation (FE)-based techniques are the standard for a wide speed range. At rotor standstill and very low speeds, High-Frequency Signal Injection (HFSI)-based rotor magnetic saliency tracking is commonly employed for state estimation [5]. However, injected signals can increase acoustic noise and degrade motor performance. Therefore, extending FE-based position extraction as close to zero speed as possible is advantageous. Furthermore, many industrial drives, such as those used in marine propulsion or pump systems, do not require continuous operation at zero speed, making them well-suited for purely FE-based techniques.

A variety of FE-based state observers, as reviewed in [3], [4], are widely utilized for rotor position estimation based on back Electromotive Force (EMF) or electromagnetic flux linkages. These observers are typically classified into three main types—Luenberger-form, Disturbance, or Filter-type—based on their mathematical equivalence [6]. They may also be categorized as full-order or reduced-order observers, depending on the rank of the observer gain matrix [7]. Additionally, concepts such as Extended EMF [8] and Active Flux (AF) [9] have been introduced to simplify the IPMSM's mathematical model, addressing complexities caused by its saliency.

A key drawback of these observers is their strong dependence on the accuracy of the identified machine parameters, often referred to as model parameters. Parameters such as $R_s$ and $\Psi_m$ are sensitive to temperature variations, while the direct and quadrature axis induc-tances ($L_d$ and $L_q$) can change due to electromagnetic saturation and cross-coupling effects. The adverse impact of inaccurate model parameters on FE-based position estimation has been well-documented [10], highlighting
issues such as poor torque and speed control, reduced efficiency, increased torque ripple, and even potential faults [11]. While full-order observers are generally more robust to parameter variations than reduced-order observers [12], incorporating online parameter adaptation can significantly improve performance [13], [14]. Additionally, online parameter tracking offers a reliable, cost-effective, and less complex alternative to sensor-based temperature monitoring by enabling the estimation of stator and rotor temperatures [15].

Consequently, online parameter adaptation has gained significant attention, with both established and emerging methods reviewed in [16], [17]. Enhancements to the Active Flux (AF) observer have been explored in [18], incorporating offline-identified $R_s$ and analytically modeled $\hat{L}_q$. In [19], the same observer is augmented with inductance and $R_s$ identification through DC voltage and signal injections, supported by additional hardware for stator voltage measurement. In [20], the full-order observer's residual information is leveraged to adapt $R_s$ and $\Psi_m$ online—targeting the low and high-speed regions, respectively. References [21]–[24] combine the Extended Electromotive Force (EEMF)-based reduced-order observer with the Recursive Least Squares (RLS) method to improve sensorless control of IPMSM. In [25], [26], signal injection techniques are used to identify multiple parameters of the PMSM sensorless drive. Additionally, modifications to the Kalman Filter for nonlinear system identification are employed in [27], [28] to estimate either or both the parameters and rotor position of the PMSM.

The authors proposed a series of recursive algorithms under the recursive prediction error method (RPEM) for PMSM identification, utilizing an open-loop predictor model and PEGs $\Psi^T$ [29], [30]. Unlike classical observers, which are designed to be insensitive to parameter variations, the open-loop model is deliberately made sensitive to parametric variations, making it more effective for identification purposes. Despite its potential, this method has not yet been explored in combination with sensorless control to assess its ability to enhance observer performance and stability. Additionally, the effective application of Hessian functions in RPEM algorithms remains unexplored. This article aims to bridge these gaps.

The main contribution of this article is the experimental validation of an open-loop predictor combined with current PEGs for sensorless drive control. The Stochastic Gradient Algorithm (SGA) is employed for online parameter identification, with its gain matrix calculated using various combinations of Hessian functions to determine optimal configurations for the simultaneous identification of $\Psi_m$ and $R_s$ in the low-speed range. Complementing the

online identification scheme, the article also introduces an offline method for identifying machine inductances using the same open-loop model. These inductances are applied in real time via a LUT, enabling online adaptation of all electrical parameters within the proposed scheme. A classical Active Flux observer is utilized for state estimation, and its performance is evaluated both with and without the integration of the parameter identification scheme, demonstrating the effectiveness of the proposed approach. Building on the authors' previous validation efforts using offline simulation [31] and real-time simulation [32], this article extends the work by presenting experimental validation using a 3 kW IPMSM and an industrial embedded control system.

## II. MACHINE MODEL AND SENSORLESS CONTROL SCHEME

This section presents the mathematical model of the IPMSM, followed by the control system including the proposed estimation schemes.

### A. IPMSM Dynamic Model

The mathematical model of the electrical part of the machine is presented in rotor coordinates and expressed in the per-unit (pu) system as follows:

$$ 
\begin{aligned}
\underline{u}_s^r &= r_s \cdot \underline{i}_s^r + \frac{1}{\omega_n} \cdot \frac{d\psi_s^r}{dt} + \mathbf{j} \cdot n \cdot \underline{\psi}_s^r \\
\underline{\psi}_s^r &= \mathbf{x}_s^r \cdot \underline{i}_s^r + \underline{\psi}_m^r
\end{aligned}
\qquad (1) $$

$$ 
\begin{aligned}
\underline{i}_s^r &= [i_d \ i_q]^T, \quad \underline{\psi}_m^r = [\psi_m \ 0]^T \\
\mathbf{x}_s^r &= \begin{bmatrix} x_d & 0 \\ 0 & x_q \end{bmatrix}, \quad \mathbf{j} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
\end{aligned}
$$

In accordance with the notations in [33], $u, i, \psi, x, n$, and $\omega_n$ represent voltage, current, flux linkage, inductances, scaled rotor speed, and nominal rotational frequency, respectively in per-unit system. The electrical angle $\vartheta$ is related to the mechanical position $\vartheta_{mech}$ as $\vartheta = p \cdot \vartheta_{mech}$, where $p$ is the number of pole pairs. To align with the notations from [34], machine parameters are denoted by $\theta$. In discrete expressions, $k$ represents the sample number in the processor, where $k = 0, 1, 2...$. Throughout the article, superscripts and subscripts denote the reference frame and the location of the quantity (stator, r-rotor, m-magnet), respectively. The symbols $\hat{}$ and superscript * represent estimated and reference quantities, respectively.
<figure><img src="image_4.png" /><figcaption>Fig. 1: Block Diagram of the vector-controlled IPMSM Sensorless Drive enhanced with the Online Parameter Estimator [30].</figcaption></figure>

## B. Overview of the IPMSM Drive

Fig. 1 illustrates the block diagram of the three-phase, current vector-controlled, sensorless IPMSM drive enhanced by a parameter identifier. The classical two-level, three-phase Voltage Source Inverter (VSI) is supplied from the DC-link, where the voltage $U_{dc}$ is measured and used to estimate the stator winding voltages. The stator current $I_s$ is measured at the output of the VSI. The parameter identifier generates the model parameter vector $\hat{\theta}$, which is fed into the state observer, along with other inputs, to estimate the rotor position $\hat{\vartheta}$ and speed $\hat{n}$. External torque or speed commands $\tau_{e,ext}^*$ and $n_{ext}^*$ can be provided from a top-level control system, such as a programmable logic controller (PLC). In speed control mode, $\tau_{e,ext}^*$ becomes redundant, as the reference torque is then generated by the speed regulator. Reference currents $i_d^*$ and $i_q^*$ are calculated to optimize either the maximum torque per ampere (MTPA) operation or a flux weakening strategy during high-speed operations.

## C. Active Flux Observer for State Estimation

$$ \psi_T = \psi_m - (x_q - x_d) \cdot i_d \quad (2a) $$

$$ \underline{\psi}_T = \underline{\psi}_s - x_q \cdot \underline{i}_s \quad (2b) $$

$$ \hat{\vartheta} = \operatorname{atan2}\left(\frac{\psi_{T,\beta}^s}{\psi_{T,\alpha}^s}\right) \quad (2c) $$

$$ \hat{n} = \frac{\psi_{T,\alpha}^s[k-1] \cdot \psi_{T,\beta}^s[k] - \psi_{T,\beta}^s[k-1] \cdot \psi_{T,\alpha}^s[k]}{T_{samp} \cdot \left(\psi_{T,\alpha}^{s2}[k] + \psi_{T,\beta}^{s2}[k]\right)} \quad (2d) $$

$$ \hat{\psi}_{s,u}^s = \omega_n \cdot \int \left( \underline{u}_s^s - r_s \cdot \underline{i}_s + \underline{u}_{comp} \right) \cdot dt \quad (3a) $$

$$ \hat{\psi}_{s,i}^r = \mathbf{x}_s^r \cdot \underline{i}_s^r + \underline{\psi}_m^r \quad (3b) $$

The classical AF observer is employed in this article for rotor position and speed estimation. The AF concept was first introduced in [9], and the design and tuning procedures for such Model Reference Adaptive System (MRAS)-type observers are detailed in [35]–[37]. For the sake of completeness, the key relevant formulae are provided in (2), where $\psi_T$ represents the AF vector.

When evaluating the impact of mistuned parameters on AF observer-based position estimation accuracy, the influence of $\hat{x}_q$ extends across the entire speed range, regardless of rotor speed, but is closely tied to the load [10]. In contrast, the effect of erroneous $\hat{x}_d$ becomes significant when the current model $\mathcal{M}_i$ dominates in the observer, which occurs primarily in the very low-speed range. During this narrow speed region, a mistuned $\hat{\psi}_m$ can also degrade position estimation accuracy. However, since the voltage model $\mathcal{M}_u$ dominates over a much wider speed range, the sensitivity to $R_s$ is most critical, particularly at speeds below mid-range [38]. This motivates the incorporation of a dedicated parameter identification scheme to support these observers, which will be presented next.

## D. Online Identification of $\psi_m$ and $r_s$ Using RPEM

RPEM for online identification of temperature-sensitive parameters is detailed in [29], [30]. For the sake of completeness, the key formulae are presented in this section.

According to [34], the general form of RPEM can be expressed as:

$$ \underline{\hat{\theta}}[k] = [\underline{\hat{\theta}}[k-1] + \mathbf{L}[k, \underline{\hat{\theta}}] \cdot \underline{\epsilon}[k, \underline{\hat{\theta}}]]_{D,\mathcal{M}} \quad (4) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2: Block diagram of the Parameter Identifier and the Online Parameter Estimator with the Open-Loop Predictor.</figcaption></figure>

Here, $\hat{\theta}$ represents the recursively estimated parameter vector, $\mathbf{L}$ denotes the estimation gain matrix, and $\epsilon$ is the prediction error, which the algorithm seeks to minimize and eventually nullify through appropriate tuning of $\mathbf{L}$. The tuning process, aimed at improving the simultaneous identification of $\hat{\psi}_m$ and $\hat{r}_s$, is detailed in Section III. The prediction error $\epsilon$ provides the RPEM algorithm with critical information about the parametric error. When $\delta\hat{\theta}$ approaches zero, the model parameter updates naturally cease.

A rank-deficiency issue [39] restricts the online identification of more than two parameters based on FE signals. To address this limitation without compromising performance, $\hat{\psi}_m$ and $\hat{r}_s$ are identified online, while $\hat{x}_d$ and $\hat{x}_q$ are determined offline and implemented in real time using a LUT. The structure of the Parameter Identifier within the proposed scheme is illustrated in Fig. 2.

$$ \frac{\hat{\mathbf{x}}_s^r(\hat{i}_s)}{\omega_n} \frac{d\hat{i}_s^r(t, \hat{\theta})}{dt} = \underline{u}_s^r(t) - \hat{r}_s(t) \cdot \hat{i}_s^r(t, \hat{\theta}) \quad (5) \\ -\mathbf{j} \cdot \hat{n}(t, \hat{\theta}) \cdot \hat{\mathbf{x}}_s^r(\hat{i}_s) \cdot \hat{i}_s^r(t, \hat{\theta}) - \mathbf{j} \cdot \hat{n}(t, \hat{\theta}) \cdot \underline{\psi}_m^r(t) $$

The predictor shown in Fig. 2 is based on the Full-Order Model $\mathcal{M}_{u\theta}$, as described in (5), which produces the estimate $\hat{i}_s^r$. The stability of the predictor, both in the continuous and discrete-time domains, is thoroughly discussed in [40], where it is concluded that the trapezoidal rule-based implementation in the DSP ensures stable prediction across the entire speed range. The prediction error, $\epsilon$, is defined as $\epsilon = \hat{i}_s^r - \hat{i}_s^r$, and a simplified expression for this error in steady-state is given in (6).

$$ \epsilon_d = - \left( \frac{\hat{n}^2 \cdot \hat{x}_q}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \right) \delta\psi_m \\ - \left( \frac{\hat{r}_s}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \cdot i_d + \frac{\hat{n} \cdot \hat{x}_q}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \cdot i_q \right) \delta r_s \\ \epsilon_q = - \left( \frac{\hat{n} \cdot \hat{r}_s}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \right) \delta\psi_m \\ - \left( \frac{\hat{r}_s}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \cdot i_q - \frac{\hat{n} \cdot \hat{x}_d}{\hat{r}_s^2 + \hat{n}^2 \cdot \hat{x}_d \cdot \hat{x}_q} \cdot i_d \right) \delta r_s \\ \delta\psi_m = \psi_m - \hat{\psi}_m, \quad \delta r_s = r_s - \hat{r}_s \quad (6) $$

The full expression that also entails $\delta x_d$ and $\delta x_q$ is disclosed in [29]. The sensitivities of $\epsilon$ in different speed and torque regions are detailed in [29], in which it is pointed out that $\epsilon$ shows less sensitivity to $\delta r_s$ beyond the low-speed region, while less sensitivity to $\delta\psi_m$ in the low-speed region. This exclusivity can be utilized in decoupled identification supplemented by a gain-scheduler.

$$ \mathbf{L}[k] = \gamma_{0,Lk} \frac{1}{r[k]} \Psi[k] \quad (7) $$

$$ r[k] = r[k-1] + \gamma_{0,rk} \left( \text{tr} \{ \Psi[k] \cdot \Psi^T[k] \} - r[k-1] \right) \quad (8) $$

$$ \Psi : \frac{d}{d\hat{\theta}} \left( \frac{d\hat{i}_s^r(t, \hat{\theta})}{dt} \right) = \frac{d}{dt} \left( \frac{d\hat{i}_s^r(t, \hat{\theta})}{d\hat{\theta}} \right) \quad (9) \\ = \frac{d}{d\hat{\theta}} \left( f_c \left( t, \hat{i}_s^r(t, \hat{\theta}), \underline{u}_s^r(t), n(t); \hat{\theta} \right) \right) $$

The gain matrix $\mathbf{L}$ can be computed using several algorithms, with the SGA proven to be the optimal choice for temperature-sensitive parameter identification [29]. The SGA is expressed in (7), where $r[k]$ represents the scalar Hessian function [34] in discrete time-domain. The dynamic or filtered form of Hessian is provided in (8). The initial value of $r[k]$ i.e., $r[0]$ must be different from zero to prevent division by zero. Additionally, $r[0]$ can be chosen to boost the convergence speed at the start of adaptation. The Hessian function is discussed in detail in Section III.

The core of the SGA is $\Psi^T$, whose generalized dynamic form is given in (9), with $f_c$ representing a function equivalent to the right-hand side of (5). The detailed investigation of PEGs is presented in [29]. Accordingly, by applying the significant PEG elements into (7) for the L-computation, (4) can be tailored to implement
online identification of temperature-sensitive parameters as follows, where $\gamma_0$ can be chosen to be a constant. [29]:

$$ \hat{\psi}_m[k] = \hat{\psi}_m[k-1] + \frac{\gamma_{0,Lk}}{r[k]} (\Psi_{11}[k] \cdot \epsilon_d[k]) \quad (10) $$

$$ \hat{r}_s[k] = \hat{r}_s[k-1] + \frac{\gamma_{0,Lk}}{r[k]} (\Psi_{21}[k] \cdot \epsilon_d[k] + \Psi_{22}[k] \cdot \epsilon_q[k]) \quad (11) $$

### E. Offline Method for $x_d$ and $x_q$ Identification

In this section, the authors propose an intuitive method for the offline identification of $\hat{x}_d$ and $\hat{x}_q$, intended for real-time implementation in the processor. The proposed approach is based on the full-order model $M u \theta$-based Open-Loop Predictor, assuming that $\hat{\psi}_m$ and $\hat{r}_s$ have been accurately identified. To minimize the impact of temperature-induced variations, tests should be conducted in short intervals at regular periods. It is crucial to ensure that there are no errors in rotor speed acquisition or rotor position offset during the experiment.

For enhanced identification accuracy, the IPMSM drive should be operated in torque mode, while the load drive operates in speed mode. In this configuration, friction and other mechanical losses are compensated by the load drive or can be neglected, allowing the stator currents in the IPMSM to be considered solely for the production of the commanded torque. If these assumptions are met and the inductances are correctly identified, the following expressions are valid in steady-state.

$$ u_d = \underbrace{r_s \cdot i_d - n \cdot x_q \cdot i_q}_{\text{Physical}} = \underbrace{\hat{r}_s \cdot \hat{i}_d - n \cdot \hat{x}_q \cdot \hat{i}_q}_{\text{Model}} $$

$$ u_q = \underbrace{r_s \cdot i_q + n \cdot x_d \cdot i_d + n \cdot \psi_m}_{\text{Physical}} = \underbrace{\hat{r}_s \cdot \hat{i}_q + n \cdot \hat{x}_d \cdot \hat{i}_d + n \cdot \hat{\psi}_m}_{\text{Model}} \quad (12) $$

Under these conditions, if discrepancies exist between the actual and predicted currents, i.e., $i_d - \hat{i}_d = \delta i_d \neq 0$ and/or $i_q - \hat{i}_q = \delta i_q \neq 0$, the following expressions can be derived by neglecting the voltage drops across the resistances, assuming that $r_s \ll x_{d,q}$.

$$ \delta i_d = \frac{u_q - n \cdot \psi_m}{n \cdot x_q \cdot \hat{x}_q} (\hat{x}_d - x_d), \quad \delta i_q = \frac{u_d}{n \cdot x_q \cdot \hat{x}_q} (x_q - \hat{x}_q) \quad (13) $$

Eq. (13) indicates that, under the given conditions, if the model inductances are accurately tuned, $\delta i_d$ and $\delta i_q$ will converge to zero. Based on this criterion, $\hat{x}_d(i_s)$ and $\hat{x}_q(i_s)$ were identified, with the experimental results shown in Fig. 3. The identified $\hat{x}_d$ shows a slight increase as $i_d$

<figure><img src="image_6.png" /><figcaption>Fig. 3: Experimentally identified $\hat{x}_d(i_s)$, $\hat{x}_q(i_s)$ using the proposed offline method.</figcaption></figure>

increases (in the negative direction). This is a possible result in IPMSM, where desaturation of the d-axis occurs due to the increase in $i_d$, which opposes the magnetic flux linkage, pulling the axis out of saturation. Additionally, mild cross-coupling effects from $i_q$ cause a slight decrease in $\hat{x}_d$ as $i_q$ increases. When considering $\hat{x}_q$, the saturation effect becomes more pronounced as $i_q$ rises. These offline identified inductances are used to develop a LUT that can be programmed in the DSP, such that the appropriate inductances can be selected in real-time.

### III. ANALYSIS AND CHOICE OF HESSIAN FUNCTIONS

#### A. Analysis of Hessian functions

$$ r_n[k] = \psi_{11}^2 + \psi_{12}^2 + \psi_{21}^2 + \psi_{22}^2 \quad (14a) $$

$$ r_{\psi m}[k] = \psi_{11}^2 + \psi_{12}^2 \quad (14b) $$

$$ r_{rs}[k] = \psi_{21}^2 + \psi_{22}^2 \quad (14c) $$

$$ r_{rd}[k] = \psi_{21}^2 \quad (14d) $$

$$ r_{rq}[k] = \psi_{22}^2 \quad (14e) $$

The Hessian function $r[k]$ and its impact on online identifications are analyzed in this section. Since $r[k]$ appears in the denominator of the L-computation, its magnitude inversely affects the gain and thus the parameter tracking speed, which, in turn, impacts the accuracy of position tracking. In accordance with [34], the expression for $r[k]$ becomes as in $r_n$, given by (14a), which is constituted of all the PEGs in L. The analysis of PEGs in [29] shows that $\psi_{11}$ is significant and consistent down to the low-speed range and rapidly decays to zero as the rotor speed approaches a standstill. $\psi_{12}$ is insignificantly small across the torque-speed plane and can thus be neglected in this analysis. In contrast, in the low-speed region, $\psi_{21}$ and $\psi_{22}$ increase rapidly as the rotor speed approaches
<figure><img src="image_7.png" /><figcaption>Fig. 4: $r_n$ behaviour in the torque-speed plane.</figcaption></figure>

a standstill. Consequently, $r_n$ yields a very large and increasing denominator as the rotor slows down to a standstill. See Fig. 4.

At very low speeds, the adaptation of $\hat{\psi}_m$ is hindered because $r_n$ reduces the estimation gain $L_{11}$ to insignificant levels, causing the adaptation process to halt. However, in the same low-speed region, the $\hat{r}_s$-adaptation remains functional as long as stator current is present. This behavior is attributed to the large values of $\psi_{21}$ and $\psi_{22}$ near zero speeds, combined with the effective noise damping provided by a substantial $r_n$ in the denominator. In summary, $r_n$ plays a critical role in naturally enabling or disabling parameter adaptations based on the respective $\epsilon$-sensitivity specific to the given speed region.

When feasible, the adaptation of $\hat{\psi}_m$ and $\hat{r}_s$ is favorable, even in overlapping speed regions. This ensures that both parameters remain accurate throughout most operating conditions, enhancing the performance of sensorless control systems. Conversely, a faulty $\hat{\psi}_m$ significantly degrades $r_s$ due to the large asymmetry in their per-unit values [29], leading to poor position estimation near zero speed. This motivates extending the adaptation speed region of $\hat{\psi}_m$ towards zero speed, which $r_n$ does not support.

Alternatively, a smaller Hessian value, such as $r_{\psi m}$ given by (14b), could be more suitable for $\hat{\psi}_m$ adaptation, as shown in Fig. 5(a), subplot-1. With $r_{\psi m}$ as the denominator, $L_{11}$ can be significantly larger at lower speeds. Beyond low speeds, $r_{\psi m}$ converges with $r_n$ due to the rapid decay of $\psi_{21,22}$-terms in $r_n$.

However, the use of $r_{\psi m}$ introduces a notable drawback: the $\epsilon$ sensitivity-based natural decoupling of $\hat{\psi}_m$-adaptation from $\delta r_s$ is no longer available. Consequently, $\hat{\psi}_m$ can be erroneously updated due to $\delta r_s$. To prevent such inaccuracies, it is necessary to either force $L_{11}$ to

<figure><img src="image_8.png" /><figcaption>Fig. 5: Behaviours of different choices for Hessians: (a) subplot-1 $r_{\psi m}$, subplot-2 $r_{rs}$ (b) subplot-1 $r_{rd}$, subplot-2 $r_{rq}$.</figcaption></figure>

zero or revert to $r_n$ in the vicinity of zero speeds.

Fig. 5(a), subplot-2 illustrates the behavior of $r_{rs}$. It closely resembles $r_n$ in the lower speed region, as $\psi_{21,22}$ dominate the full trace at these speeds. Alternatively, $r_{rd}$ and $r_{rq}$ (Fig. 5(b)) can be used as Hessians for $L_{21}$ and $L_{22}$ computations, respectively. However, $r_{rd}$ exhibits a dip at $n = 0.01$ pu, causing Hessian to be zero. Care must be taken to avoid division by zero and excessive noise amplification. In summary, the Hessians play a crucial role in adjusting convergence speeds, particularly during the simultaneous adaptation of both $\hat{\psi}_m$ and $\hat{r}_s$, as explored in the next section.

## B. Hessians' Influence on Simultaneous Adaptation

While $\epsilon$ with respect to $\hat{\psi}_m$ shows strong sensitivity only beyond the low-speed region, it exhibits moderate sensitivity in the low-speed region, which overlaps with the high-sensitivity region for $\epsilon$ with respect to $\hat{r}_s$. Fig. 6 illustrates these sensitivity regions and the gain-scheduler proposed in [30]. A well-designed gain matrix enables

<figure><img src="image_9.png" /><figcaption>Fig. 6: Gain-scheduling scheme to prevent undue $\hat{r}_s$-adaptation in the speeds beyond $|n_{lim}|$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7: Simultaneous adaptation of $\hat{\psi}_m$ and $\hat{r}_s$ when both are erroneous by 10% at n = 0.05 pu, $\tau_{el} = 0.5$: (a) when SGA is formulated using $r[k] = r_n[k]$ vs. when using (14b) and (14c) for $\hat{\psi}_m$ and $\hat{r}_s$ respectively,</figcaption></figure>

simultaneous adaptation of both $\hat{\psi}_m$ and $\hat{r}_s$ within this region.

In this investigation, the simultaneous tracking behavior of $\hat{\psi}_m$ and $\hat{r}_s$ is evaluated based on various Hessian solutions presented in (14). The evaluation is conducted using a SoC-based real-time simulation platform, as described in [41], to perform dynamic simulations.

Fig. 7 and Fig. 8 illustrate the simulation results of simultaneous adaptation at 0.05 pu speed and 0.5 load torque. In Fig. 7, the results are presented for two cases. The gain matrix is formulated using either (14a) or (14b) for $\hat{\psi}_m$, and for $\hat{r}_s$ adaptation, either (14a) or (14c) applied. It is observed that, in contrast to the full trace, the traces with selective prediction gradients result in faster tracking. Consequently, the position error $\delta\theta$ converges towards zero more quickly with selective traces. This phenomenon is further emphasized in Fig. 8. Here, the trace for $\hat{r}_s$-tracking is limited to $r[k] = r_{rq}[k]$, which further boosts the $\hat{r}_s$ estimation gains. Consequently, $\hat{\psi}_m$-tracking also becomes faster. In both figures, it is evident that using selective prediction gradients provides faster and more efficient adaptation compared to using the full trace with all prediction gradients.

According to the findings, the stochastic gradient algorithms with gain matrix optimized for simultaneous

<figure><img src="image_11.png" /><figcaption>Fig. 8: Simultaneous adaptation of $\hat{\psi}_m$ and $\hat{r}_s$ when both are erroneous by 10% at n = 0.05 pu, $\tau_{el} = 0.5$: (a) when SGA is formulated using $r[k] = r_n[k]$ vs. when using (14b) and (14c) for $\hat{\psi}_m$ and $\hat{r}_s$ respectively.</figcaption></figure>

online identification of $\hat{\psi}_m$ and $\hat{r}_s$ become:

$$ \hat{\psi}_m[k] = \hat{\psi}_m[k-1] + \frac{\gamma_{0,Lk}}{r_{\psi m}[k]} (\Psi_{11}[k] \cdot \epsilon_d[k]) \quad (15) $$

$$ \hat{r}_s[k] = \hat{r}_s[k-1] + \frac{\gamma_{0,Lk}}{r_{rq}[k]} (\Psi_{22}[k] \cdot \epsilon_q[k]) \quad (16) $$

## IV. EXPERIMENTAL VALIDATION

This section presents the results and discussion of the experimental validation. The experiments were conducted using a 3 kW IPMSM laboratory prototype, as shown in Fig. 9. The IPMSM is a three-phase, six-pole, 36-slot machine with single-layer stator winding. The machine's ratings and electrical parameters are summarized in Table I. The drive software and firmware are implemented on the Xilinx Zynq SoC, housed in the PicoZed System-on-Module. The modulators and digital filters are programmed within the FPGA fabric of the SoC. The proposed estimation algorithms and slower processes are programmed in the on-chip processor using the C++ programming language.

The IGBT switching frequency ($f_{sw}$) is set at 4 kHz, with the processor sampling time ($T_{ samp}$) synchronized to the PWM double-update sampling step. A two-level VSI with asymmetrical modulation and third harmonic injection is used to drive the machine. The offline identified inductance-current mappings from Fig. 3 are used
<table><caption>TABLE I: IPMSM PARAMETERS</caption><thead><tr><th>Symbol</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>U<sub>n</sub></td><td>Rated Voltage</td><td>400 V</td></tr><tr><td>I<sub>n</sub></td><td>Rated Current</td><td>4.93 A</td></tr><tr><td>P<sub>n</sub></td><td>Rated Power</td><td>3 kW</td></tr><tr><td>N<sub>n</sub></td><td>Rated Speed</td><td>1000 rpm</td></tr><tr><td>T<sub>n</sub></td><td>Rated Torque</td><td>32.6 Nm</td></tr><tr><td>p</td><td>Number of pole-pairs</td><td>3</td></tr><tr><td>R<sub>s</sub></td><td>Stator Resistance (offline)</td><td>2.25 Ω</td></tr><tr><td>Ψ<sub>m</sub></td><td>Permanent magnet flux linkage (offline)</td><td>1.14 Wb</td></tr><tr><td>L<sub>d</sub></td><td>d-axis inductance (rated load)</td><td>95.3 mH</td></tr><tr><td>L<sub>q</sub></td><td>q-axis inductance (rated load)</td><td>206 mH</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Fig. 9: Experimental Setup having IPMSM as the drive machine and DC machine as the load machine mounted on the same shaft.</figcaption></figure>

through a LUT for real-time updating of $\hat{x}_d$ and $\hat{x}_q$. Compensation for VSI nonlinearity and dead-time effects is applied following the method in [42].

A. Steady-State Performance

Figs. 5 and 6 illustrate different test cases where the drive operates in mechanical steady-state, while initially off-tuned electrical parameters are subsequently tuned online. Figs. 10(a) and 10(b) show the performance at very low speeds of 0.01 and 0.02 pu, under load torques $\tau_{e,l}$ of 0.5 and 0.8 pu, respectively. Initially, $\hat{r}_s$ was incorrectly set, but at the 0-second mark, the online adaptation scheme was activated. At 0.01 pu speed, the improvement in $\delta\vartheta$ is significant, reducing from approximately 20° to nearly 0°, owing to the accurate online adaptation of $\hat{r}_s$ and accurate computation of $\hat{x}_q$. Fig. 10(c) further demonstrates the performance before and after $\hat{r}_s$ adaptation at 0.05 pu speed, where the improvement in $\delta\vartheta$ is clearly visible. Fig. 5(d) highlights the performance enhancement at 0.3 pu speed, when $\hat{\psi}_m$ is identified online. In this case, $\delta\vartheta$ remains unaffected, as the observer relies solely on $\mathcal{M}_u$.

Similarly, Figs. 11(a), (b), and (c) demonstrate the improvements the active flux observer gains through the

<figure><img src="image_13.png" /><figcaption>Fig. 10: Experimental validations of the sensorless IPMSM drive before and after enabling the $\hat{r}_s$ adaptation (a) with $\tau_{e,l} = 0.5$ pu, $n^* = 0.01$ pu (b) with $\tau_{e,l} = 0.8$ pu, $n^* = 0.02$ pu (c) with $\tau_{e,l} = 0.5$ pu, $n^* = 0.05$ pu (d) $\hat{\psi}_m$ adaptation with $\tau_{e,l} = 0.8$ pu, $n^* = 0.3$ pu.</figcaption></figure>

updating of $\hat{x}_q$, compared to its nameplate value $\hat{x}_{q0}$. Likewise, Fig. 11(d) shows the effect of updating $\hat{x}_d$ in contrast to its nameplate value $\hat{x}_{d0}$. During no-load conditions (Fig. 11(a)), updating $\hat{x}_q$ has no impact on position estimation since $\hat{\psi}_T = \hat{\psi}_s$. However, in subsequent cases where $\hat{i}_s \neq 0$, the improvement in state estimation is evident. Similarly, the $\hat{x}_d$-updating also shows no impact
<figure><img src="image_14.png" /><figcaption>Fig. 11: Experimental validations of the sensorless IPMSM drive before and after enabling the online updating of the $\hat{x}_q$ (a) with $\tau_{e,l} = 0$ pu, $n^* = 0.3$ pu (b) with $\tau_{e,l} = 0.4$ pu, $n^* = 0.1$ pu (c) with $\tau_{e,l} = 0.8$ pu, $n^* = 0.02$ pu (d) $\hat{x}_d$ updating with $\tau_{e,l} = 0.8$ pu, $n^* = 0.02$ pu.</figcaption></figure>

on position estimation accuracy, as seen in the final figure. Overall, these results successfully validate the proposed offline experiment method for identifying $x_d$ and $x_q$.

## B. Dynamic Performance

The performance and stability of the proposed scheme under dynamic conditions are evaluated in this section. Figs. 12(a) and 12(b) show speed reversal tests at no load and with 0.5 load torque. In both cases, position estimation remains excellent across the entire speed range, except during zero crossing, where $\delta\vartheta$ momentarily increases. However, once the back EMF signal becomes strong enough, the observer quickly recovers, providing accurate estimations. These figures also display the traces of all relevant electrical parameters, showing that both $\hat{\psi}_m$ and $\hat{r}_s$ accurately track their physical counterparts, even in the speed-region that enables simultaneous adaptation. Figs. 12(c) and 12(d) depict the system's response to a speed step and a load-torque step, respectively. Overall, the proposed sensorless scheme demonstrates acceptable performance and stability under dynamic mechanical conditions.

<figure><img src="image_15.png" /><figcaption>Fig. 12: Dynamic performance of the sensorless IPMSM drive: speed reversal (a) from 0.35 to -0.35 pu at $\tau_{e,l} = 0$ pu (b) speed reversal from -0.3 to +0.3 pu at $\tau_{e,l} = 0.5$ pu (c) speed step change from 0.1 to 0.3 pu at $\tau_{e,l} = 0.5$ pu (d) Load-torque step change from 0.4 to 0.6 pu at $n^* = 0.2$ pu.</figcaption></figure>

## C. Summary of Validation Results

The overall position estimation accuracy of the proposed combined scheme is summarized in Table II. The results demonstrate commendable accuracy in position estimation, which is attributed to the real-time updating of
<table><caption>TABLE II: Active flux observer performance comparison with and without the online parameter updating in different operating regions.</caption><thead><tr><th rowspan="2">Operating range</th><th colspan="2">Max. δθ</th></tr><tr><th>No updating</th><th>With updating</th></tr></thead><tbody><tr><td colspan="3"><strong>When δr<sub>s</sub> = 75 - 100%</strong></td></tr><tr><td>At 0.01 - 0.05 pu speed range</td><td>&gt; 15°</td><td>&lt; 2°</td></tr><tr><td>At 0.05 - 0.1 pu speed range</td><td>~ 2°</td><td>&lt; 0.5°</td></tr><tr><td>At 0.1 and above</td><td>&gt; 1°</td><td>~ 0°</td></tr><tr><td colspan="3"><strong>When δx<sub>q</sub> = ~5%</strong></td></tr><tr><td>At full load</td><td>~ 2°</td><td>~ 0°</td></tr><tr><td><strong>Processor execution speed</strong></td><td>~5 µs</td><td>~9.5 µs</td></tr></tbody></table>

the IPMSM electrical parameters. The contribution from $\hat{x}_q$-updating is noticeable across the entire speed range, while $\hat{r}_s$-adaptation proves especially beneficial at lower speeds, as expected. Both $\hat{\psi}_m$ and $\hat{r}_s$ converge within a few seconds, regardless of speed or load torque $\tau_{e,l}$. This convergence speed is acceptable, given the large thermal capacity that prevents sudden changes in these temperature-sensitive parameters. In contrast, inductance updates occur with only a single sampling delay $T_{samp}$.

The execution time of the proposed combined scheme on the ARM processor is approximately 9 µs, with position estimation alone taking around 4 µs. For most industrial drives employing IGBT-based inverters, the additional processing time introduced by the Parameter Identifier can be conveniently accommodated, unless the application demands the use of low-cost processors or operates at very high rotor speeds.

## V. CONCLUSION

This article proposed a method to improve the sensorless control of three-phase IPMSM drives by integrating a parameter adaptation scheme with an open-loop current predictor. The inherent sensitivity of the open-loop predictor to model parameters was utilized by employing prediction error gradient-based SGA to compute the estimation gains in real-time. Different Hessian function options were explored using a real-time simulation platform to tune the estimation gain matrix for optimal simultaneous online identification of $R_s$ and $\Psi_m$. Additionally, the paper introduced an offline method to experimentally identify $\hat{L}_d$ and $\hat{L}_q$, using the same open-loop predictor. During experimental validation, $R_s$ and $\Psi_m$ were identified online, while the offline-identified $\hat{L}_d$ and $\hat{L}_q$ were updated online using a programmed LUT in the processor. The proposed sensorless scheme significantly enhanced position estimation, particularly in the very low-speed range, even under varying parameters and load conditions. The results demonstrated that sustained low-

speed operation is feasible without the need for test signal injection. Consequently, this approach offers a promising solution for a wide range of industrial applications where continuous operation near rotor standstill is not required.

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