# Angular position estimation error extraction for speed and angular position estimation of IPMSM using a parameter-free adaptive observer

Enrique Alvaro-Mendoza, Jesus De-Leon Morales, Mohamed Assaad Hamida, Malek Ghanes

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## To cite this version:

Enrique Alvaro-Mendoza, Jesus De-Leon Morales, Mohamed Assaad Hamida, Malek Ghanes. Angular position estimation error extraction for speed and angular position estimation of IPMSM using a parameter-free adaptive observer. Journal of The Franklin Institute, 2022, 359 (13), pp.7140-7164. 10.1016/j.jfranklin.2022.06.021. hal-03894370

HAL Id: hal-03894370

https://hal.science/hal-03894370v1

Submitted on 28 May 2025

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# Angular position estimation error extraction for speed and angular position estimation of IPMSM using a parameter-free adaptive observer

Enrique Alvaro-Mendoza<sup>a,b</sup>, Jesús De León-Morales<sup>a</sup>, Mohamed Assaad Hamida<sup>b</sup>, Malek Ghanes<sup>b</sup>

## Abstract

In this paper, an alternative approach for extracting the angular position estimation error for an Interior Permanent Magnet Synchronous Machine (IPMSM) is introduced. The originality of this work consists on obtaining the angular position estimation error by the measurable currents, in the $\alpha\beta$ stationary reference frame, without considering the high frequency signal injection. Then, using the obtained angular position estimation error, an Adaptive High Order Sliding Mode Observer (AHOSMO) is proposed for estimating the position, speed and acceleration of the IPMSM. The observer is used for a parameter-free virtual dynamical system to avoid impact of the parameter variations on the estimation. The convergence analysis of the proposed observer is established using a Lyapunov stability theory. Furthermore, simulation and experimental results are presented to illustrate the efficiency of the proposed approach against disturbances and noises. The proposed approach is compared with high frequency (HF) and back electromotive force (BEMF) approaches.

**Keywords:** Sensorless, Parameter independent, Sliding mode, IPMSM, Experimental results

## 1. Introduction

Interior Permanent Magnet Synchronous Machine (IPMSM) is generally used in many engineering applications, such as electric vehicles, intelligent robotics, aerospace, etc., thanks to its interesting advantages such as high efficiency, high torque capability, small size, high-power density and wide-speed range operations.

In engineering applications, speed and angular position information is required to control the IPMSM. Conventional control systems use encoders to obtain information about the angular position, which reduce reliability, increase noise, require maintenance and increase the cost of the system. Therefore, in order to overcome this problem, an alternative is the sensorless technique, such that, information about the angular position and speed is obtained from measurable voltages and currents.

Preprint submitted to Journal of LATEX Templates

May 20, 2022

<footnote><sup>a</sup>Facultad de ingenieria Mecanica y Electrica, Universidad Autónoma de Nuevo León, San Nicolas De Los Garza, 66451, Nuevo León, México</footnote>

<footnote><sup>b</sup>Ecole Centrale de Nantes, LS2N UMR CNRS 6004, Nantes, France</footnote>
There exist two main approaches to give solutions to the sensorless technique: Saliency-based methods and model-based methods.

Saliency-based methods are dependent on machine inductance and have a focus on inductance variation property by high frequency signal injection. It is worth mentioning that an enhancement for sensorless control has been achieved with the high-frequency injection signal due to that rotor position can be estimated at low speeds and even at standstill operation. Several techniques have been proposed in [1, 2, 3, 4]. However, saliency-based methods present some problems, such as acoustic noises and torque ripples, reason why these methods are sometimes limited in real applications. Moreover, a persistent excitation condition must be satisfied by the high-frequency injection signal; otherwise, the angular position is not observable.

On the other hand, model-based methods have been widely applied using dynamical models of the IPMSM to design observers and estimate the angular position and speed. The main advantage of these methods is that the rotor position estimation has an efficient performance at medium and high speed. The model-based methods have been proposed using Extended Kalman filter [5, 6, 7, 8, 9], Luenberger observer [10, 11], sliding mode observer [12, 13, 14, 15], etc. Several approaches have been presented in [16, 17, 18, 19, 20]. Nevertheless, the estimation based on models can fail at low and zero speed.

In last years, sliding mode observers have been extensively used in sensorless applications due to their strong robustness with respect to disturbances and uncertainties. However, the conventional sliding mode observer has chattering problem, which represents its major drawback. Several approaches to diminish this undesirable phenomenon have been addressed. For instance, in [21], a sigmoid function has replaced the sign function to avoid chattering. Similarly, low-pass filters have been used to reduce the chattering, which generate a phase delay in the system, such that it is necessary to implement another strategy to overcome this problem [22]. However, high order sliding mode approach has been the better alternative to greatly mitigate the chattering problem [23].

On the other side, it is well known that the machine parameters vary significantly depending on operation conditions. The electrical parameters could vary depending on the temperature variations, magnetic circuit saturation and so on; and the mechanical machine parameters could vary significantly according to the applied load torque, weight, road type and tires quality in automotive applications. Because of that, proposing a parameter independent estimation strategy is an urgent task. Then, in this work, two principal contributions are proposed:

1) A novel sensorless strategy for IPMSM, which combines saliency and model-based methods for the extraction of the angular position estimation error $e_{\theta} = \theta - \hat{\theta}$ is proposed. The angular position estimation error $e_{\theta}$ is determined in terms of the measurable currents in a $\alpha\beta$ stationary reference frame. Unlike other methods where the angular position estimation error is obtained from back electromotive force (BEMF) or using high-frequency injection, the angular position estimation error is independent of all machine parameters and high frequency signal injection characteristics. Therefore, this improves the feasibility
of design and reduce the cost of the implementation.

2) By using the rotor position estimation error $e_\theta$ and based on virtual system, an Adaptive High Order Sliding Mode Observer (AHOSMO) is proposed to estimate the rotor position, the speed and the acceleration of IPMSM. The designed observer is independent from the machine parameters and requires only one parameter for its tuning. As well as the robustness the practical implementation of the proposed observer is easier compared to existing ones.

A convergence analysis of the estimation error using Lyapunov stability theory is addressed. Moreover, sufficient conditions are derived to ensure the convergence in finite time of the estimation error and obtain a better robustness and fast convergence in the presence of disturbances and noises.

The paper is organized as follows: In section 2, problem statement is introduced. In section 3, a new strategy for obtaining the angular position estimation error is addressed. Then, an AHOSMO and its convergence analysis are introduced in section 4. In section 5, simulation and experimental results are introduced to demonstrate the effectiveness of proposed observer. Finally, a conclusion is given in section 6.

## 2. Problem statement

Frequently, in several industrial applications, the control of IPMSM requires the knowledge of the angular position and speed, which usually are not available by measurement. Then one solution is to propose strategies to estimate the angular position and speed. These strategies are often based on model. However, the parametric uncertainties and external disturbances affect the behaviour of the system and they must be considered in the estimation. One solution to overcome this drawback is the robust observer design to estimate the angular position and speed of the IPMSM. Then, the most frequently used mathematical model of the IPMSM for control and observer design, is the model represented by the interconnection of the electrical and mechanical sub-systems, in a dq synchronous reference frame [23], i.e.

$$ \Sigma_{elec}: \left\{ \begin{aligned} \frac{di_d}{dt} &= -\frac{R_s}{L_d}i_d + p\Omega\frac{L_q}{L_d}i_q + \frac{v_d}{L_d} \\ \frac{di_q}{dt} &= -\frac{R_s}{L_q}i_q - p\Omega\frac{L_d}{L_q}i_d + \frac{v_q}{L_q} - p\Omega\frac{\psi_r}{L_q} \end{aligned} \right. \quad (1) $$

$$ \Sigma_{mech}: \left\{ \begin{aligned} \frac{d\theta}{dt} &= \Omega. \\ \frac{d\Omega}{dt} &= \frac{p}{J}(L_d - L_q)i_d i_q + \frac{p}{J}\psi_r i_q - \frac{f_v}{J}\Omega - \frac{1}{J}T_l \end{aligned} \right. \quad (2) $$

where the electrical sub-system (1) is represented by: $i_d$ and $i_q$, which are the dq-axes stator currents, $v_d$ and $v_q$ are the dq-axes stator voltages, $R_s$ is the stator resistance, $L_d$ and $L_q$ are the dq-axes winding inductance and $\psi_r$ is the permanent-magnet flux linkage.
Furthermore, the mechanical sub-system (2) is represented by: the angular position $\theta$ and the speed $\Omega$, where, $J$ is the moment of inertia; $f_v$ is the viscous friction coefficient; $p$ is the number of poles pairs and $T_l$ the load torque.

However, in order to estimate the angular position and overcome the issues caused by the parametric uncertainties present in the model of the IPMSM, in this work, an observer based on an angular position estimation error $e_\theta$ extraction approach is considered, using a parameter-free virtual system. Then, a different strategy has been proposed by considering only the measurable signals of the IPMSM, i.e., the currents $i_\alpha$ and $i_\beta$, which can be obtained from the abc triphasic components of the IPMSM and can be used for extracting the angular position estimation error $e_\theta$.

Notice that, in order to overcome the use of the mathematical model of the IPMSM depending on parameters; the angular position, speed and acceleration will be estimated by the following parameter free virtual system

$$ 
\begin{aligned}
\frac{d\theta}{dt} &= \Omega \\
\frac{d\Omega}{dt} &= \alpha \\
\frac{d\alpha}{dt} &= \rho(t)
\end{aligned}
\qquad (3) $$

where $\theta$ is the angular position, $\Omega$ is the speed and $\alpha$ is the acceleration, where the time derivative of the acceleration is equal $\rho(t)$, which is an unknown and bounded function depending on the angular position, speed, currents and parameters.

It is clear that the mechanical sub-system (2) of the IPMSM does not depend on the acceleration, however, to estimate the position and the speed, the mechanical sub-system has been extended including the acceleration in order to improve the estimation of those variables. In other words, the first two equations of (3) are enough to have a good estimation with low transient modes. However, with fast dynamics, speed estimation errors could increase due to that its derivative is supposed to be equal to zero. Therefore, to overcome this problem, the machine acceleration-$\alpha$ is also estimated to have a more precise estimation in transient mode. Therefore, the virtual system (3) will be used to estimate angular position, speed and acceleration by using an extraction of the angular position estimation error $e_\theta$.

### 3. Angular position estimation error extraction

Now, a procedure to extract the angular position estimation error $e_\theta$ of IPMSM, from available currents, in a $\alpha\beta$-reference frame, will be introduced.

Consider Park transformation given by

$$ \mathbb{P}(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \qquad (4) $$
and the measurable currents $i_\alpha$ and $i_\beta$, which can be expressed in terms of the currents $i_d$ and $i_q$, by the following equations

$$ \begin{bmatrix} i_\alpha \\ i_\beta \end{bmatrix} = \mathbb{P}(\theta) \begin{bmatrix} i_d \\ i_q \end{bmatrix} \qquad (5) $$

where

$$ \begin{aligned} i_\alpha &= \cos(\theta)i_d - \sin(\theta)i_q \\ i_\beta &= \sin(\theta)i_d + \cos(\theta)i_q \end{aligned} \qquad (6) $$

Notice that the currents $i_\alpha$ and $i_\beta$ depend on the angular position and the currents $i_d$ and $i_q$. However, the information of these variables are not available by measurement.

Then, using the following transformation matrix $T_\theta$

$$ T_\theta = \begin{bmatrix} \cos(\hat{\theta} + \phi) & \sin(\hat{\theta} + \phi) \\ -\sin(\hat{\theta} + \phi) & \cos(\hat{\theta} + \phi) \end{bmatrix} \qquad (7) $$

where $\hat{\theta}$ is estimated angular position shifted by $\phi$. Then, choosing $\phi$ appropriately, the angular position error can be extracted.

Therefore, using the transformation matrix $T_\theta$, the currents $i_\alpha$ and $i_\beta$ can be transformed, into alternate synchronous reference frame, as follows

$$ \begin{bmatrix} I_{d\theta} \\ I_{q\theta} \end{bmatrix} = T_{\theta} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} \qquad (8) $$

where

$$ \begin{aligned} I_{d\theta} &= i_{\alpha} \cos(\hat{\theta} + \phi) + i_{\beta} \sin(\hat{\theta} + \phi) \\ &= \cos(\theta - \hat{\theta} - \phi)i_{d} - \sin(\theta - \hat{\theta} - \phi)i_{q} \end{aligned} \qquad (9) \\ I_{q\theta} &= -i_{\alpha} \sin(\hat{\theta} + \phi) + i_{\beta} \cos(\hat{\theta} + \phi) \\ &= \sin(\theta - \hat{\theta} - \phi)i_{d} + \cos(\theta - \hat{\theta} - \phi)i_{q} \end{aligned} $$

Now, taking into account some ideas of the saliency-based method [24], and without considering the high-frequency injection signal, and under the action of a controller, such that $i_d$ tends to a reference $i_d^*$. Then, to extract the angular position error, the following equation is defined as follows

$$ E_{\theta} = I_{q\theta} - I_{d\theta} + i_{d}^{*}\sqrt{2} \qquad (10) $$

Notice that $E_\theta = E_\theta(i_\alpha, i_\beta, i_d^*, \phi, \hat{\theta})$ can be calculated from $(i_\alpha, i_\beta, i_d^*, \phi)$, which are known values and $\hat{\theta}$ will be computed by using the observer introduced later.

Now, in order to write $E_\theta$ in terms of the angular position estimation error $e_\theta = \theta - \hat{\theta}$, we proceed as follows. Choosing $\phi = \frac{\pi}{4}$, then $E_\theta$ can be expressed, in terms of $i_d, i_q$ and the angular position estimation error $e_\theta = \theta - \hat{\theta}$, as follows

$$ E_{\theta} = i_{q}\sqrt{2} \sin(e_{\theta}) - i_{d}\sqrt{2} \cos(e_{\theta}) + i_{d}^{*}\sqrt{2} \qquad (11) $$
Therefore, assuming that a controller is applied such that the current $-i_d$ tracks the reference current $-i_d^*$; then, the following equation is obtained

$$E_{\theta} = i_{q} \sqrt{2} \sin(e_{\theta}) + i_{d}^{*} \sqrt{2} [1 - \cos(e_{\theta})] \quad (12)$$

or equivalently

$$E_{\theta} = i_{q} \sqrt{2} \sin(e_{\theta}) + i_{d}^{*} \sqrt{2} \left[ 2 \left( \sin\left(\frac{e_{\theta}}{2}\right) \right)^{2} \right] \quad (13)$$

Therefore, taking into account a small angular position estimation error $e_{\theta}$, $E_{\theta}$ can be approximated, in terms of $e_{\theta}$ explicitly, as follows

$$E_{\theta} \approx i_{q} \sqrt{2}(e_{\theta}) + \frac{i_{d}^{*}}{\sqrt{2}}(e_{\theta}^{2}) \quad (14)$$

Taking into account that quadratic term is sufficiently small with respect to linear term, then, $E_{\theta}$ can be expressed as

$$E_{\theta} \approx i_{q} \sqrt{2} (e_{\theta}) \quad (15)$$

Notice that (15) depends of the current $i_q$. It is worth mentioning that the changes in the current $i_q$ are directly proportional to the electromagnetic torque $T_e$ [see [25]]. As can be seen in Fig.1, different profiles of speed and electromagnetic torque have been plotted and the behaviors of the currents $i_d$ and $i_q$ are shown. Then, from Fig.1, current $i_q$ can be positive

<figure><img src="image_3.png" /><figcaption>Figure 1: Speed and electromagnetic torque profiles and behaviour of currents $i_d$, $i_q$.</figcaption></figure>

or negative depending on the electrical machine operation. Then, multiplying $sign(i_q)$ in both side of the equation, it follows that

$$\begin{aligned} E_{\theta} \operatorname{sign}(i_q) &\approx i_q \operatorname{sign}(i_q) \sqrt{2} (e_{\theta}) \\ &\approx \sqrt{2} |i_q| e_{\theta} \end{aligned} \quad (16)$$

Taking into account that $i_{qmax} > |i_q|$. Finally, it follows that

$$\tilde{E}_{\theta} \approx \gamma e_{\theta} \quad (17)$$
with $\tilde{E}_{\theta} = E_{\theta}\text{sign}(i_q)$, and $\gamma = i_{qmax}\sqrt{2}$, where $i_{qmax}$ is the maximum value of $i_q$.

Nevertheless, $i_q$ is not available for measurement. Then, in (17), $i_q$ will be replaced by the estimated current $\hat{i}_q$. In fact, $\hat{i}_q$ is obtained from the transformation of the measured currents $i_{\alpha}$ and $i_{\beta}$ from the stator reference frame to synchronous reference frame as shown by these equations

$$ \begin{bmatrix} \hat{i}_d \\ \hat{i}_q \end{bmatrix} = \mathbb{P}(\hat{\theta})^{-1} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} \qquad (18) $$

where $\mathbb{P}(\hat{\theta}) = \begin{bmatrix} \cos(\hat{\theta}) & -\sin(\hat{\theta}) \\ \sin(\hat{\theta}) & \cos(\hat{\theta}) \end{bmatrix}$ and since $\hat{\theta}$ will be calculated by the proposed observer, then,

$$ \hat{i}_q = -\sin(\hat{\theta})i_{\alpha} + \cos(\hat{\theta})i_{\beta} \qquad (19) $$

The initial rotor position information is needed for practical implementation to obtain $\hat{i}_q$. This problem is addressed in the literature by several research works (as it could be seen in [26, 27, 28]) and is supposed to be solved. In the experimental implementation, the rotor is moved very slightly by applying short voltage in order to detect the initial rotor position information. Once this information is obtained, the current $\hat{i}_q$ could be calculated using equation (19). Based on the calculated $\hat{i}_q$, the rotor position estimation error could be extracted by

$$ e_{\theta} \approx \frac{\tilde{E}_{\theta}}{\gamma} = \frac{E_{\theta}\text{sign}(\hat{i}_{q})}{\gamma} \qquad (20) $$

where $E_{\theta}$ is computed from (10). In the sequel, (20) will be used in the AHOSMO for estimating the angular position, speed and acceleration.

## 4. Adaptive observer design and convergence analysis

Consider the following class of nonlinear system

$$ \begin{aligned} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= x_3 \\ \dot{x}_3 &= \rho(t) \\ y &= x_1 \end{aligned} \qquad (21) $$

where $x_1$, $x_2$ and $x_3$ represent the states, $y$ the output and $\rho(t)$ is an unknown and bounded term of the system. Then, from the system (21), an AHOSMO is designed and its analysis of convergence is introduced.

**Assumption A.** The term $\rho(t)$ is unknown and bounded, i.e., $|\rho(t)| \le \varrho_1$, for $\varrho_1 > 0$.
Then, an AHOSMO for the system (21) is given by

$$
\begin{align}
\dot{x}_1 &= \hat{x}_2 + \lambda_1 |x_1 - \hat{x}_1|^{\frac{2}{3}} \mathrm{sign}(x_1 - \hat{x}_1) \nonumber \\
\dot{x}_2 &= \hat{x}_3 + \lambda_2 |x_1 - \hat{x}_1|^{\frac{1}{3}} \mathrm{sign}(x_1 - \hat{x}_1) \nonumber \\
\dot{x}_3 &= \lambda_3 \mathrm{sign}(x_1 - \hat{x}_1) \nonumber \\
\hat{y} &= \hat{x}_1 \tag{22}
\end{align}
$$

where $\hat{x}_1$, $\hat{x}_2$ and $\hat{x}_3$ represent the estimated states and $\hat{y}$ the estimated output of the system.
Furthermore, the gains are given by

$$
\lambda_1 = 3L \quad \lambda_2 = 2L^2 \quad \lambda_3 = \left(\frac{2}{3}\right)^2 L^3 \qquad (23)
$$

where $L > 0$ is an adaptive gain that will be introduced later. On the other hand, the state estimation errors are defined as follows

180

$$
e_1 = x_1 - \hat{x}_1, \quad e_2 = x_2 - \hat{x}_2, \quad e_3 = x_3 - \hat{x}_3 \tag{24}
$$

and the dynamics of the estimation errors are given by

$$
\begin{align}
\dot{e}_1 &= e_2 - \lambda_1 |e_1|^{\frac{2}{3}} \mathrm{sign}(e_1) \notag \\
\dot{e}_2 &= e_3 - \lambda_2 |e_1|^{\frac{1}{3}} \mathrm{sign}(e_1) \tag{25} \\
\dot{e}_3 &= -\lambda_3 \mathrm{sign}(e_1) + \rho(t) \notag
\end{align}
$$

Now, in order to analyse the convergence of the observer, a change of variable is intro-
duced

$$
\ell_1 = \frac{|e_1|^{\frac{2}{3}} \mathrm{sign}(e_1)}{L}, \quad \ell_2 = \frac{e_2}{L^2}, \quad \ell_3 = \frac{3e_3 |e_1|^{\frac{1}{3}}}{2L^3} \quad (26)
$$

such that the dynamics of (26) are given by

$$
\begin{align}
\dot{\ell}_1 &= \frac{2L}{3|e_1|^{\frac{1}{3}}} [-3\ell_1 + \ell_2] - \frac{\dot{L}}{L}\ell_1 \nonumber \\
\dot{\ell}_1 &= \frac{2L}{3|e_1|^{\frac{1}{3}}} [-3\ell_1 + \ell_3] - 2\frac{\dot{L}}{L}\ell_2 \tag{27} \\
\dot{\ell}_3 &= \frac{2L}{3|e_1|^{\frac{1}{3}}} \left[ -\ell_1 + \left(\frac{3}{2}\right)^2 \frac{|e_1|^{\frac{2}{3}}}{L^4} \rho(t) + \frac{L\ell_3}{2|e_1|^{\frac{2}{3}}} [-3\ell_1 + \ell_2] \right] - 3\frac{\dot{L}}{L}\ell_3 \nonumber
\end{align}
$$

185

Then, in order to reduce the calculations later, the system (27) can be rewritten in a
compact form as follows

$$
\dot{\ell} = \alpha \left[ (A - S_x^{-1} C^T C) \ell + \Phi \right] - \frac{\dot{L}}{L} D \ell \quad (28)
$$
where $\alpha = \frac{2L}{3|e_1|^{\frac{1}{3}}}$ and

$$ \mathcal{L} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix}, \quad A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \quad S_x = \begin{bmatrix} 1 & -1 & 1 \\ -1 & 2 & -3 \\ 1 & -3 & 6 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}, $$

$$ D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}, \quad \Phi = \begin{bmatrix} 0 \\ 0 \\ \left(\frac{3}{2}\right)^2 \frac{|e_1|^2}{L^4} \rho(t) + \frac{L \mathcal{L}_3}{2|e_1|^2} [-3\mathcal{L}_1 + \mathcal{L}_2] \end{bmatrix}. $$

**Assumption B.** The terms in the vector $\Phi$ are uniformly bounded with respect to $u$ and locally Lipschitz with respect to $\mathcal{L}$, i.e., $||\Phi|| \le \hbar||\mathcal{L}||$, for $\hbar > 0$.

Moreover, $S_x$ can be defined as a symmetric positive-definite matrix and its solution is given by

$$ S_x + A^T S_x + S_x A - C^T C = 0 \quad (29) $$

**Theorem 1.** Consider the dynamic system (21) and the Assumptions A and B are satisfied. Furthermore,

$$ \dot{L} = \left[ k^{\frac{1}{2}} |e_1|^{\frac{2}{3}} - \mu^{\frac{1}{2}} L^2 \right] \quad (30) $$

is an adaptive law of $L$, for $\mu > 0$ and $k > 0$ chosen appropriately. Then, the system (22) is an Adaptive High Order Sliding Mode Observer (AHOSMO) for the dynamic system (21), more precisely, estimation errors $e_i$, for $i = 1, 2, 3$; converge to zero in finite time.

**Proof:**

Consider a strong Lyapunov function as follows

$$ V_{(\mathcal{L},L)} = V_{(\mathcal{L})} + V_{(L)} \quad (31) $$

defining $V_{(\mathcal{L})} = \mathcal{L}^T S_x \mathcal{L}$ and $V_{(L)} = \frac{\mu}{2} L^2$. Then, considering the suitable expressions, the first derivative in time of the Lyapunov function is given by

$$ \dot{V}_{(\mathcal{L},L)} = \alpha \mathcal{L}^T [A^T S_x + S_x A] \mathcal{L} - 2\alpha \mathcal{L}^T C^T C \mathcal{L} - 2 \frac{\dot{L}}{L} \mathcal{L}^T S_x D \mathcal{L} + \mu \dot{L} L + 2\alpha \mathcal{L}^T S_x \Phi \quad (32) $$

Taking into account that $A^T S_x + S_x A = -S_x + C^T C$. Equation (32) can be rewritten as follows

$$ \dot{V}_{(\mathcal{L},L)} = -\alpha \mathcal{L}^T S_x \mathcal{L} - \alpha \mathcal{L}^T C^T C \mathcal{L} - 2 \frac{\dot{L}}{L} \mathcal{L}^T S_x D \mathcal{L} + \mu \dot{L} L + 2\alpha \mathcal{L}^T S_x \Phi \quad (33) $$

On the other hand, considering that the inequalities,

$$ \frac{|e_1|^4}{L^2} = |\mathcal{L}_1|^2 \le ||\mathcal{L}||^2 \quad (34) $$
and

$$ \lambda_{min}(S_x)||\mathcal{L}||^2 \leq V_{(\mathcal{L})} \leq \lambda_{max}(S_x)||\mathcal{L}||^2, \quad (35) $$

205 are fulfilled, where $\lambda_{min}(S_x)$ and $\lambda_{max}(S_x)$ are the minimum and maximum singular values of $S_x$. Moreover,

$$ \lambda_{min}(S_x D)||\mathcal{L}||^2 \leq \mathcal{L}^T S_x D \mathcal{L} \leq \lambda_{max}(S_x D)||\mathcal{L}||^2 \quad (36) $$

where $\lambda_{min}(S_x D)$ and $\lambda_{max}(S_x D)$ are the minimum and maximum singular values of $S_x D$. Then,

$$ \dot{V}_{(\mathcal{L},L)} \leq -\alpha \mathcal{L}^T S_x \mathcal{L} - \alpha \mathcal{L}^T C^T C \mathcal{L} - 2\lambda_{min}(S_x D) ||\mathcal{L}||^2 \frac{\dot{L}}{L} + \mu \dot{L} L + 2\alpha \mathcal{L}^T S_x \Phi \quad (37) $$

such that,

$$ \dot{V}_{(\mathcal{L},L)} \leq -\alpha \mathcal{L}^T S_x \mathcal{L} - \alpha \mathcal{L}^T C^T C \mathcal{L} - k ||\mathcal{L}||^2 \frac{\dot{L}}{L} + \mu \dot{L} L + 2\alpha \mathcal{L}^T S_x \Phi \quad (38) $$

210 where $k = 2\lambda_{min}(S_x D) > 0$. Thus, the above equation can be written as follows

$$ \dot{V}_{(\mathcal{L},L)} \leq -\alpha \mathcal{L}^T S_x \mathcal{L} - \alpha \mathcal{L}^T C^T C \mathcal{L} + 2\alpha \mathcal{L}^T S_x \Phi - \frac{\dot{L}}{L} [k ||\mathcal{L}||^2 - \mu L^2] \quad (39) $$

and the last term of (39) can be factored as follows

$$ \frac{\dot{L}}{L} [k ||\mathcal{L}||^2 - \mu L^2] = \frac{\dot{L}}{L} [k^{\frac{1}{2}} ||\mathcal{L}|| + \mu^{\frac{1}{2}} L] [k^{\frac{1}{2}} ||\mathcal{L}|| - \mu^{\frac{1}{2}} L] \quad (40) $$

defining $f_{(\mathcal{L},L)} = [k^{\frac{1}{2}} ||\mathcal{L}|| + \mu^{\frac{1}{2}} L] > 0$. Then,

$$ \dot{V}_{(\mathcal{L},L)} \leq -\alpha \mathcal{L}^T S_x \mathcal{L} - \alpha \mathcal{L}^T C^T C \mathcal{L} + 2\alpha \mathcal{L}^T S_x \Phi - f_{(\mathcal{L},L)} \frac{\dot{L}}{L} [k^{\frac{1}{2}} ||\mathcal{L}|| - \mu^{\frac{1}{2}} L] \quad (41) $$

Now, from inequalities (34) and (35), the following inequality is satisfied

$$ \frac{|e_1|^{\frac{2}{3}}}{L} \leq ||\mathcal{L}|| \leq \left( \frac{V_{(\mathcal{L})}}{\lambda_{min}(S_x)} \right)^{\frac{1}{2}} \quad (42) $$

Then, from (42), it follows that

$$ \dot{V}_{(\mathcal{L},L)} \leq -\alpha V_{(\mathcal{L})} - \alpha \mathcal{L}^T C^T C \mathcal{L} + 2\alpha \mathcal{L}^T S_x \Phi - f_{(\mathcal{L},L)} \frac{\dot{L}}{L} \left[ k^{\frac{1}{2}} \frac{|e_1|^{\frac{2}{3}}}{L} - \mu^{\frac{1}{2}} L \right] \quad (43) $$

215 Therefore, choosing an adaptive law as follows

$$ \dot{L} = \left[ k^{\frac{1}{2}} \frac{|e_1|^{\frac{2}{3}}}{L} - \mu^{\frac{1}{2}} L \right] L \quad (44) $$
Then,

$$ \dot{V}_{(\ell,L)} \leq -\alpha V_{(\ell)} - \alpha \ell^T C^T C \ell - f_{(\ell,L)} \left[ k^{\frac{1}{2}} \frac{|e_1|^2}{L} - \mu^{\frac{1}{2}} L \right]^2 + 2\alpha \ell^T S_x \Phi \quad (45) $$

Assuming that $e_1$ tend to zero faster than $L$, and the term $(-\alpha \ell^T C^T C \ell) < 0$, for $L > 0$. Equation (45) is given by

$$ \dot{V}_{(\ell,L)} \leq -\alpha V_{(\ell)} - f_{(\ell,L)} [\mu L^2] + 2\alpha \ell^T S_x \Phi \quad (46) $$

Moreover, taking the norm to the nonlinear term $2\alpha \ell^T S_x \Phi$ and from Assumption B, then (46) is given by

$$ \dot{V}_{(\ell,L)} \leq -\alpha V_{(\ell)} + 2\alpha ||\ell||^2 ||S_x|| \ell - f_{(\ell,L)} [\mu L^2] \quad (47) $$

Now, from inequality (35), it follows that

$$ \dot{V}_{(\ell,L)} \leq - \frac{2L}{3|e_1|^{\frac{1}{3}}} [1 - \sigma] V_{(\ell)} - f_{(\ell,L)} [\mu L^2] \quad (48) $$

where $\sigma = \frac{2||S_x||\ell}{\lambda_{max}(S_x)}$. Furthermore, from $\frac{|e_1|^2}{L} = |\ell_1| \leq |\ell_1|^2 \leq ||\ell||^2$, the following inequality is satisfied

$$ \frac{|e_1|^{\frac{1}{3}}}{L^{\frac{1}{2}}} \leq ||\ell|| \leq \left( \frac{V_{(\ell)}}{\lambda_{min}(S_x)} \right)^{\frac{1}{2}} \quad (49) $$

and considering that (48) can be written as follows

$$ \dot{V}_{(\ell,L)} \leq - \frac{2L}{\frac{3|e_1|^{\frac{1}{3}} L^{\frac{1}{2}}}{L^{\frac{1}{2}}}} [1 - \sigma] V_{(\ell)} - f_{(\ell,L)} [\mu L^2] \quad (50) $$

Then,

$$ \dot{V}_{(\ell,L)} \leq -L^{\frac{1}{2}} \Gamma V_{(\ell)}^{\frac{1}{2}} - f_{(\ell,L)} [\mu L^2] \quad (51) $$

where $\Gamma = \frac{2[1 - \sigma]\lambda_{min}^{\frac{1}{2}}(S_x)}{3}$. Rewritten (51) as follows

$$ \dot{V}_{(\ell,L)} \leq -L\sqrt{2}\mu^{\frac{1}{2}} \left[ \frac{\Gamma}{\sqrt{2}\mu^{\frac{1}{2}}L^{\frac{1}{2}}} V_{(\ell)}^{\frac{1}{2}} + f_{(\ell,L)} \frac{\mu^{\frac{1}{2}}}{\sqrt{2}} L \right] \quad (52) $$

and defining $\eta_1 = L\sqrt{2}\mu^{\frac{1}{2}}$ and $\varphi = min(\frac{\Gamma}{\sqrt{2}\mu^{\frac{1}{2}}L^{\frac{1}{2}}}, f_{(\ell,L)})$. The following equation can be expressed as

$$ \dot{V}_{(\ell,L)} \leq -\eta \left[ V_{(\ell)}^{\frac{1}{2}} + \frac{\mu^{\frac{1}{2}}}{\sqrt{2}} L \right] \quad (53) $$
where $\eta = \eta_1\varphi$. On the other hand, considering that Jensen's inequality is defined as follows

$$[|a|^q + |b|^q]^{\frac{1}{q}} \le |a| + |b| \qquad (54)$$

and defining $a = V_{(\mathcal{L})}^{\frac{1}{2}}$, $b = V_{(L)}^{\frac{1}{2}}$ and $q = 2$. Then, the following inequality is fulfilled

$$\left[ |V_{(\mathcal{L})}^{\frac{1}{2}}|^2 + |V_{(L)}^{\frac{1}{2}}|^2 \right]^{\frac{1}{2}} \le |V_{(\mathcal{L})}^{\frac{1}{2}}| + \left| \frac{\mu^{\frac{1}{2}}}{\sqrt{2}} L \right| \qquad (55)$$

such that

$$V_{(\mathcal{L},L)}^{\frac{1}{2}} \le V_{(\mathcal{L})}^{\frac{1}{2}} + \frac{\mu^{\frac{1}{2}}}{\sqrt{2}} L \qquad (56)$$

Finally, the Lyapunov dynamic equation is satisfied as follows

$$\dot{V}_{(\mathcal{L},L)} \le -\eta V_{(\mathcal{L},L)}^{\frac{1}{2}} \qquad (57)$$

As mentioned before, $\dot{V}_{(\mathcal{L},L)}$ is a strong Lyapunov function, with $L$ sufficiently large, satisfying $\eta > 0$. Then, $\dot{V}_{(\mathcal{L},L)}$ is negative definite and can guarantee the convergence of the observer in finite time. On the other side, taking into account the equation $\dot{v} = -\eta v^{\frac{1}{2}}$, whose solution is defined by $v(t) = (v(0))^{\frac{1}{2}} - \frac{1}{2}\eta t)^2$. Then, the comparison principle can be applied in order to estimate the convergence time $T$. Therefore, $V_{(\mathcal{L},L)} < v(t)$ when $V_{(\mathcal{L}(0)),L(0))} < v(0)$, then $\mathcal{L}$ has a finite-time convergence in an estimated time defined by $T = \frac{2V_{(\mathcal{L}(0)),L(0))}{}$ for $L$ sufficiently large. Thus, $V_{(\mathcal{L},L)}$ tends to zero in finite-time, which involves that the estimation errors $e_i$, for $i = 1, 2, 3$; tend to zero in finite time.

**Remark 1.** As can be seen, the system (28) has a singularity when $e_1 = 0$. The singularity arise due to the change of variable $\mathcal{L}_1 = \frac{|e_1|^2}{L} \text{sign}(e_1)$, $\mathcal{L}_2 = \frac{e_2}{L^2}$, $\mathcal{L}_3 = \frac{3e_3|e_1|^{\frac{1}{3}}}{2L^3}$ converting system (24) into system (28), whose domain is defined as follows $\mathcal{D} = \{(\mathcal{L}_1, \mathcal{L}_2, \mathcal{L}_3) \in \mathbb{R}^3 | \mathcal{L}_1 \neq 0\}$. Nonetheless, considering convergence analysis, the singularity does not appear when the system is expressed in terms of the original coordinates (see for more details [29, 30]).

### 4.1. AHOSMO design for the IPMSM

From Theorem 1, consider the following AHOSMO for the virtual system (3), it is given by

$$\begin{aligned}
\dot{\hat{\theta}} &= \hat{\Omega} + \lambda_1 |\theta - \hat{\theta}|^{\frac{2}{3}} \text{sign}(\theta - \hat{\theta}) \\
\dot{\hat{\Omega}} &= \alpha + \lambda_2 |\theta - \hat{\theta}|^{\frac{1}{3}} \text{sign}(\theta - \hat{\theta}) \\
\dot{\hat{\alpha}} &= \lambda_3 \text{sign}(\theta - \hat{\theta})
\end{aligned} \qquad (58)$$

where $\hat{\theta}$, $\hat{\Omega}$ and $\hat{\alpha}$ are the estimated of the angular position, speed and acceleration, respectively; and $\lambda_1$, $\lambda_2$ and $\lambda_3$ are the gains defined in (23). However, $e_\theta = \theta - \hat{\theta}$ is not available
for measurement, which difficult its implementation. Then, to overcome this difficulty, a strategy for extracting $e_θ$ was introduced in section 3. Therefore, since $\tilde{E}_θ$ is known by measurement of the currents $i_{αβ}$ and is expressed as $\tilde{E}_θ = γe_θ$ for $γ > 0$. Then, $(θ - \hat{θ})$ will be replaced by $(θ - \hat{θ}) = \frac{\tilde{E}_θ}{γ}$ in the observer (58), i.e.

$$ 
\begin{aligned}
\dot{\hat{\theta}} &= \hat{\Omega} + \lambda_1 \left| \frac{\tilde{E}_\theta}{\gamma} \right|^{\frac{2}{3}} \mathrm{sign}\left(\frac{\tilde{E}_\theta}{\gamma}\right) \\
\dot{\hat{\Omega}} &= \hat{\alpha} + \lambda_2 \left| \frac{\tilde{E}_\theta}{\gamma} \right|^{\frac{1}{3}} \mathrm{sign}\left(\frac{\tilde{E}_\theta}{\gamma}\right) \\
\dot{\hat{\alpha}} &= \lambda_3 \mathrm{sign}\left(\frac{\tilde{E}_\theta}{\gamma}\right)
\end{aligned}
\qquad (59) $$

which is implemented to estimate the angular position, speed and acceleration. The pro-

<figure><img src="image_4.png" /><figcaption>Figure 2: Proposed observer scheme</figcaption></figure>

posed strategy is summarized in Fig. 2 in order to schematically show the interconnection of the observer.

## 5. Simulation and experimental results

Simulation and experimental results have been evaluated in order to show the performance of the proposed strategy. The parameters of the IPMSM are presented in the Table 1. Moreover, the parameters used in the AHOSMO are the following: In simulation test $k = 130$ and $μ = 0.002$; and in experimental test $k = 80$ and $μ = 0.002$ with an initial condition $L(0) = 30$.

### 5.1. Hardware Description

The experimental setup is shown by Fig. 3 and is composed by an IPMSM rated at 3 kW supplied by a three-phase voltages source inverter is arranged for the experimental test.
<table><caption>Table 1: IPMSM nominal parameters</caption><thead><tr><td>Symbol</td><td>Parameter</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>Stator resistance</td><td>1.4</td><td>ohms</td></tr><tr><td>J</td><td>Moment of inertia</td><td>7.3e<sup>-3</sup></td><td>kg.m<sup>2</sup></td></tr><tr><td>p</td><td>Number of poles</td><td>5</td><td></td></tr><tr><td>T<sub>l</sub></td><td>Torque</td><td>9</td><td>N-m</td></tr><tr><td>ψ<sub>r</sub></td><td>Permanent-magnet flux linkage</td><td>0.18</td><td>Wb</td></tr><tr><td>L<sub>d</sub></td><td>d-axis winding inductance</td><td>0.0057</td><td>H</td></tr><tr><td>L<sub>q</sub></td><td>q-axis winding inductance</td><td>0.0099</td><td>H</td></tr><tr><td>f<sub>v</sub></td><td>Viscous friction coefficient</td><td>0.0034</td><td>kg-m<sup>2</sup>/s</td></tr></tbody></table>

An inverter composed by insulated gate bipolar transistor (IGBTs) as power switches has been used, which is powered by 400 V dc voltage. The pulse width modulation (PWM) technique is generated by a dSPACE DS1103 with a switching frequency of 10 kHz. The digital board of dSPACE receives the stator currents and the dc link voltages data with a 10 kHz frequency, and the measured torque data with a 2 kHz frequency. The load torque is generated by a PMSM mechanically coupled with the shaft of the IPMSM. The angular position is measured by an encoder. Moreover, a Kalman-filter applied to the measured position is used to calculate the rotor speed data.

<figure><img src="image_5.png" /><figcaption>Figure 3: Experimental setup</figcaption></figure>

Furthermore, in order to show the performance of the proposed strategy, a comparative study has been carried out. The proposed AHOSMO is compared with a BEMF-based observer and a HF injection-based observer from simulation and experimentation.

## 5.2. Benchmark Cycle

The profile of speed considered in simulation and experimentation is shown in Fig.4, where the speed is 0 rad/s from 0 s to 0.5 s, after that, the speed increases of 0 rad/s to 314 rad/s from 0.5 s to 3 s, from this time the speed is 314 rad/s until 3.5 s, after that, the speed decreases of 314 rad/s to −314 rad/s from 3.5 s to 4.5 s; during 0.5 s the speed is −314 rad/s; after that, the speed tends to −150 rad/s from 5 s to 6 s and stays in this
speed for 0.5 s, after that, the speed tends to 0 rad/s from 6.5 s to 7.5 s and stays in this speed until 10 s. Moreover, the load torque applied has a value of 4 N.m from 0 s to 3.3 s, after that, the load torque decrease to -4 N.m from 3.3 s to 4.7 s and stays there until 10 s.

290

<figure><img src="image_6.png" /><figcaption>Figure 4: Profile of speed and load torque applied for simulation and experimentation</figcaption></figure>

Moreover, in order to validate the performance of the proposed AHOSMO based on the extraction of angular position estimation error $e_\theta$, an experimental test is carried out taking into account a low speed region and different profiles of small load torque as can be seen in Fig. 5. An explanation about performance will be introduced later.

295

On the other hand, the electrical parameters (resistance and inductance) could vary depend-

<figure><img src="image_7.png" /><figcaption>Figure 5: Profiles of speed and load torque considering a low-speed region and different small load torque profiles.</figcaption></figure>

300

ing on the temperature variations, magnetic circuit saturation and so on; and the mechanical machine parameters (inertia and viscous coefficient), could vary significantly according to the applied load torque, weight, road type and tires quality in automotive applications. However, since that it is not possible to have access to machine parameters experimentally, the experimental tests are made during a large space of time in order to see the effect of the parameters on the proposed strategy. Moreover, a parameter variations test is made in simulation by considering parameter profiles as shown by Fig. 6, where variations in the resistance and inductance have been considered in order to show the effectiveness of the AHOSMO under variations in the parameters.

305
<figure><img src="image_8.png" /><figcaption>Figure 6: Parameter variations in simulation results</figcaption></figure>

### 5.3. *Simulation tests*

In simulations, two tests have been evaluated, the first test is carried out by considering the parameters of the Table 1 and the second test is carried out considering parametric uncertainties, as shown in Fig.6.

As previously mentioned, the proposed AHOSMO is compared with a BEMF-based observer and a HF injection-based observer from simulation tests. First, an observer for the BEMF is implemented for estimating the angular position and the speed, using nominal parameters of the IPMSM, as can be seen in Fig. 7. In addition, parameter variations have been applied as shown in Fig.6 in order to evaluate the robustness of this scheme, notice that a low performance is obtained as shown in Fig.8.

In the BEMF-based observer, angular position is estimated by $arctan(-\hat{e}_{\alpha}/\hat{e}_{\beta})$, where $\hat{e}_{\alpha}$ and $\hat{e}_{\beta}$ are the estimated BEMF in a stationary $\alpha\beta$-frame. However, if speed is calculated from the estimated angular position time derivative, the noise affects the estimation. Then, a Low Pass Filter (LPF) is used to mitigate this problem. However, it introduces a phase delay in the system. Moreover, BEMF-based observer is dependent on time variant parameters.

On the other hand, a HF injection-based observer is considered for estimating angular position and speed, and results using nominal parameters are plotted in Fig. 9, where HF injection-based observer performs better than BEMF-based observer.

However, HF injection-based observer always requires HF injection and filters to achieve a signal conditioning. Simulation results have been plotted under inductance variations, which can be seen in Fig.10. Notice that HF injection-based observer is sensitive to inductance variations.

Now, the proposed AHOSMO using nominal parameters is implemented to estimate angular position, speed and acceleration as can be shown in Fig. 11. Notice that proposed AHOSMO performs better than the other strategies due to AHOSMO does not require additional elements for achieving the estimation, such as filters or HF injection. Furthermore, the AHOSMO is robust under parameter uncertainties at low and high speeds, it can be seen in Fig.12. Then, the action of the adaptive law L can be seen in Fig. 13 considering
<figure><img src="image_9.png" /><figcaption>Figure 7: Simulation test: BEMF-based observer in nominal case</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: Simulation test: BEMF-based observer with parameter variations</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 9: Simulation test: HF injection-based observer in nominal case</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10: Simulation test: HF injection-based observer with parameter variations</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 11: Simulation test: Proposed AHOSMO in nominal case</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: Simulation test: Proposed AHOSMO with parameter variations</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 13: Simulation test: Behaviour of the adaptive law considering two cycles of the speed reference</figcaption></figure>

two cycles of the speed reference in order to show its behaviour during the state estimation.

## 5.4. *Experimental tests*

In the same way as it was carried out in simulation, the proposed AHOSMO is compared with a BEMF-based observer and a HF injection-based observer from experimental tests. The experimental tests have been carried out by considering the nominal parameters of the IPMSM in the comparison techniques in order to make a fair comparison due to that the proposed observer does not use parameters. However, as previously mentioned, the machine parameters may vary during experiments due to several effects such as the temperature and the magnetic saturation and so on. In Fig.14, the BEMF-based observer presents a large estimation error at low speed. It is well known this strategy has observability problems at low speed.

On the other hand, the estimations of speed and angular position using HF injection-based observer are shown by Fig.15, where a good performance is obtained. However, the AHOSMO performs better than HF injection-based observer as can be seen in Fig.16.

In Fig. 17a the action of the adaptive law $L$ is introduced, considering two cycles of the reference speed as can be seen in Fig.17b in order to show its behaviour. Then, better results have been obtained by using an adaptive gain.

Now, in Table 2, results of the comparative study are introduced, where the effectiveness of the proposed AHOSMO is highlighted. The performance indices taken into account in the comparative study are Integral absolute error (IAE) and integral square error (ISE). Then, Table 2 can show that the proposed AHOSMO has a significant improvement with respect to the others strategies.

On the other hand, the performance of the proposed observer based on the extraction of $e_{\theta}$ is evaluated in an experimental test considering the profiles of the Fig. 5. A low-speed and zero region is taken into account due to that in this region most of the observers present observability problems. It is well known that the IPMSM is not observable when the angular speed is equal to zero. However, in the proposed strategy, the angular position estimation error $e_{\theta}$ extracted depends on the dynamics of the current-$i_q$ directly. Therefore, the observability is ensured for a current-$i_q$ different to zero, i.e., $i_q \neq 0$, such that this condition is satisfied when the load torque or the speed are different to zero. In this way,
<figure><img src="image_16.png" /><figcaption>Figure 14: Experimental test: BEMF-based observer</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 15: Experimental test: HF injection-based observer</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 16: Experimental test: Proposed AHOSMO</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 17: Experimental test: Behaviour of the adaptive law considering two cycles of the speed reference</figcaption></figure>

the load torque profile considered in the validation has values equals to zero and different to zero with small values.

Therefore, from Fig. 18, it is shown that at the beginning, the speed is 0 rad/s with a load torque going from 0.05 N.m to 1 N.m. Then, the observer converges to real angular position and speed. After that, from 1.5 s to 4s the load torque is 0 N.m and the speed is still 0 rad/s until 3 s. Therefore, from 1.5 s, the observer diverges, since, there exist a loss of observability when both speed and load torque are zero, since at that moment the electric machine is standstill and there is not a persistent current—$i_q$ in $\tilde{E}_\theta$. However, from 3 s the speed increase until 2.5 rad/s, such that, the current—$i_q$ is different to 0, then the observer
<table><caption>Table 2: Comparative study summary</caption><thead><tr><th colspan="4">Performance indices from simulation results with parameters variations</th></tr><tr><th></th><th></th><th>Proposed strategy</th><th>HF-injection</th><th>BEMF</th></tr></thead><tbody><tr><td rowspan="2">-Speed estimation error</td><td>IAE</td><td>0.088</td><td>5.5285</td><td>43.0487</td></tr><tr><td>ISE</td><td>0.0128</td><td>5.7241</td><td>476.8356</td></tr><tr><td rowspan="2">-Angular position estimation error</td><td>IAE</td><td>8.4099e<sup>-5</sup></td><td>1.2056</td><td>5.1706</td></tr><tr><td>ISE</td><td>1.7067e<sup>-6</sup></td><td>0.2177</td><td>6.1884</td></tr></tbody></table><table><thead><tr><th colspan="4">Performance indices from experimental results</th></tr><tr><th></th><th></th><th>Proposed strategy</th><th>HF-injection</th><th>BEMF</th></tr></thead><tbody><tr><td rowspan="2">-Speed estimation error</td><td>IAE</td><td>9.1457</td><td>14.9466</td><td>43.7955</td></tr><tr><td>ISE</td><td>18.0503</td><td>60.6974</td><td>623.4936</td></tr><tr><td rowspan="2">-Angular position estimation error</td><td>IAE</td><td>0.1749</td><td>1.2488</td><td>8.5005</td></tr><tr><td>ISE</td><td>0.5120</td><td>3.5251</td><td>25.908</td></tr></tbody></table><table><thead><tr><th>Parameter dependency</th><th>Not</th><th>Yes</th><th>Yes</th></tr></thead></table>

tends towards real speed and angular position. Therefore, from 3.5 s is possible to see the convergence in the angular position. Then, the speed stays at low speed (2.5 rad/s) for 2 s with a load torque different to 0 N.m, such that, a good estimation is achieved for the observer. After that, the speed increases until 30 rad/s and stays there for 3 s, and the load torque tends to 0 N.m and stays there from 8.5 s to 11 s, such that, the observer achieves a very accurate estimate. Finally, the speed decreases until 1.8 rad/s and from 13 s until 16 s, the speed continues to decrease until it reaches 0 rad/s with a small load torque of 0.05 N.m. From this test it can be concluded that the observability depends directly on $i_q$ which must be different from zero ($i_q \neq 0$) to ensure the observability.

## 6. Conclusion

A novel approach for obtaining the angular position estimation error has been developed in this paper based on the measured currents in the $\alpha\beta$ stationary reference frame without high frequency signal injection. Then, to estimate the rotor position, speed and acceleration based on the extracted error $e_\theta$, an Adaptive High Order Sliding Mode observer (AHOSMO) has been introduced for a class of nonlinear system. This observer is used for a parameter-free virtual mechanical system to overcome the problem of parametric uncertainties in the IPMSM. The developed strategy is tested in simulation under electrical and mechanical parameters profiles introduced to test the robustness of the system. The obtained results are very satisfactory and have shown that the estimation is not affected by the parameter variations. Compared experimentally with back EMF and HF signal injection methods, the validity and superiority of the proposed algorithm are verified through experiments in different situations.
<figure><img src="image_20.png" /><figcaption>Figure 18: Experimental Result: Convergence of proposed AHOSMO and behaviour of current $i_q$, applying different profiles of small load Torque and considering a low-speed region.</figcaption></figure>

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