Adaptive QSMO-Based Sensorless Drive for IPM Motor with NN-Based Transient Position Error Compensation

Abstract

In commercial electrical equipment, the popular sensorless drive scheme for the interior permanent magnet synchronous motor, based on the quasi-sliding mode observer (QSMO) and phase-locked loop (PLL), still faces challenges such as position errors and limited applicability across a wide speed range. To address these problems, this paper analyzes the frequency domain model of the QSMO. A QSMO-based parameter adaptation method is proposed to adjust the boundary layer and widen the speed operating range, considering the QSMO bandwidth. A QSMO-based phase lag compensation method is proposed to mitigate steady-state position errors, considering the QSMO phase lag. Then, the PLL model is analyzed to select the estimated speed difference for transient position error compensation. Specifically, a transient position error compensator based on a feedback time delay neural network (FB-TDNN) is proposed. Based on the back propagation learning algorithm, the specific structure and optimal parameters of the FB-TDNN are determined during the offline training process. The proposed parameter adaptation method and two position error compensation methods were validated through simulations in simulated wide-speed operation scenarios, including sudden speed changes. Overall, the proposed scheme fully mitigates steady-state position errors, substantially mitigates transient position errors, and exhibits good stability across a wide speed range.

1. Introduction

Interior permanent magnet (IPM) synchronous motors have characteristics such as wide speed range operation, high reliability, and high efficiency [1,2]. Hence, IPM motors are widely used in various commercial electrical equipment applications, such as electric vehicles, household appliances, and industrial automation, etc. In these applications, rotor position and rotational speed information is essential for field-oriented control (FOC), coordinate transformation, and speed closed-loop tracking. Rotor position information needs to be obtained through mechanical position sensors. These sensors, such as encoders, resolvers, and Hall sensors, function well from medium to high speeds in these roles. However, the installation of mechanical position sensors raises costs [3,4], complicates compact product design [5], and compromises motor robustness in harsh conditions [6,7]. To overcome these limitations, researchers have developed various sensorless drive techniques. These techniques enable accurate position and speed estimation without position sensors. Meanwhile, these solutions effectively mitigate the drawbacks associated with sensor deployment and achieve performance comparable to position sensor-based drives.

Currently, popular sensorless techniques are primarily developed from two perspectives: fundamental frequency motor models and observers [8]. From a model perspective, sensorless drive methods can be designed based on either flux linkage or back electromotive force (EMF) [9,10,11,12,13,14,15]. In flux linkage-based methods, the rotor flux linkage containing position information can be estimated from the stator flux linkage. The stator flux linkage is derived from the stator voltage model of IPM motors [9]. These methods have good performance theoretically when the motor operates at medium to high speeds in the considered applications [9,10]. However, the integral term in the stator flux linkage presents challenges, such as determining the initial conditions for the integral and addressing drift in flux linkage-based methods [9,11,12]. Meanwhile, the presence of the integral term also means that these methods heavily rely on the quality and accuracy of voltage and current measurements [8,12].

EMF-based methods, which are also suitable for medium- to high-speed applications, effectively circumvent these challenging issues. However, these methods are only applicable to surface-mounted permanent magnet (SPM) synchronous motors [13,14,15]. The saliency effect of IPM motors results in the inclusion of rotor position not only in the back EMF term but also in the inductance term. To address this issue, researchers have reconstructed the back EMF into the extended EMF (EEMF). This approach has successfully estimated the position and speed of IPM motors [16,17,18], as only the EEMF term contains position information. Therefore, the motor model with EEMF is considered in the proposed sensorless drive scheme design in this paper. Subsequently, it is crucial to select an appropriate observer to effectively implement the EEMF-based drive scheme.

Mainstream observer methods for sensorless drives include the Luenberger observer, disturbance observer, extended Kalman filter (EKF), and sliding mode observer (SMO). Linear Luenberger observers, relying on first-order Taylor expansion, are effective but limited in a broad state-space region. To widen this region, advanced nonlinear versions incorporate a nonlinear correction term for accurate estimation [19,20]. However, observer parameters must align strictly with actual motor parameters, which may not always remain nominal [19]. Disturbance observers estimate the EEMF by appropriately configuring observer gains and treating the EEMF as a disturbance. The authors in [16,21] achieved position estimation in the $dq$ and $\alpha \beta$ coordinate systems, respectively. However, similar to Luenberger observers, variations in motor parameters still affect the accuracy of position estimation [8]. Hence, these two types of methods may not be suitable for mass-produced electrical equipment that only provides motor nominal values.

As a possible alternative, EKFs exhibit robustness to parameter variations and measurement noise due to their stochastic properties [22,23]. Nevertheless, their strong robustness to measurement noise also presents challenges. It is difficult to adjust the complex covariance matrices of measurement noise and models in these methods [22]. SMOs are also robust to parameter variations due to their variable structure characteristics [24]. Specifically, a signum switching function of estimated current errors steers the operating point to the sliding manifold. This enables real-time estimation of back EMF or EEMF. Compared with EKFs, SMOs are easier to implement and have lower computational demands. This has made SMOs widely applicable in sensorless drives using discrete-time microcontrollers, such as DSPs [13,14].

However, in discrete-time systems, the discontinuity and high-frequency switching of the signum function can cause chattering or oscillation issues [25]. Generally, to mitigate chattering, a low-pass filter (LPF) is integrated into the SMO structure. This introduces unwanted phase lag, which necessitates additional position compensation. To address chattering issues fundamentally, the continuous-time SMO can be replaced with a discrete-time or quasi-SMO (QSMO). In a QSMO, a continuous switching function, such as a saturation function or sigmoid function, is used instead of the steep signum function. It provides smooth transition switching around the sliding manifold. The authors in [7,15] employed QSMOs to successfully estimate the EEMF and EMF, without incorporating an LPF within the control structure. Subsequently, position and speed information can be simultaneously extracted from the back EMF or EEMF using a phase-locked loop (PLL) [26]. Compared to arctan-based extraction methods, a PLL can also achieve non-delayed position filtering without the need for an additional LPF [26,27]. Finally, a QSMO and a PLL are selected to design the proposed sensorless drive for an IPM motor in this paper.

To achieve adaptive sensorless drive over a wide speed range, the research undertaken in [7,28] constrained the discrete state trajectory of the QSMO operating point. It can effectively mitigate chattering across various speeds through boundary layer adaptive adjustment. However, the analysis process from a discrete-time perspective is overly complex. In [29], the shape coefficient, which is also the boundary layer of continuous switching functions, was determined by a trial-and-error method. Nonetheless, a fixed boundary layer proves effective only within a limited speed range. In [15], the frequency domain equivalent model of the QSMO is proposed solely for compensating position deviation in an SPM motor. The QSMO model exhibits a low-pass characteristic akin to an LPF, highlighting its potential for effective chattering filtering. Hence, the consideration of the QSMO bandwidth provides a reliable basis for the QSMO-based parameter adaptation method proposed in this paper. Moreover, to mitigate steady-state position errors caused by QSMO, a QSMO-based phase lag compensation method similar to that in [15] is considered in this paper.

However, the sensorless drive scheme with a PLL still exhibits position errors in scenarios of rapid speed variations. These errors can be reduced by increasing the bandwidth of the PLL. However, this approach heightens sensitivity to disturbances, potentially destabilizing the motor drive system. Therefore, the PLL bandwidth needs to be kept at a lower level. To address this problem, researchers have proposed various position error compensation methods [15,30,31,32]. In [15,30,31], speed feedforward compensation methods with speed LPFs were proposed to mitigate position errors during acceleration and deceleration. In [32], an enhanced PLL with a double integrator was proposed to achieve zero position error. However, the methods mentioned above require well-designed filters and proficient frequency domain analysis skills. Fortunately, artificial intelligence technology exhibits powerful nonlinear fitting and estimation capabilities [33]. Hence, this technology has introduced numerous new techniques and attractive implementations to the field of motor drives. Among these, the feedback time delay neural network (FB-TDNN) is notable for its excellent capabilities in long-term dependency learning and time series data modeling [34]. Therefore, in this paper, an offline FB-TDNN is selected to design a transient position error compensator.

The major contributions of this paper are the proposal of an adaptive QSMO-based sensorless drive scheme, a steady-state position error mitigation method, and a transient position error compensation method for an IPM motor. Compared to the conventional QSMO-based scheme, the proposed scheme enables stable operation of an IPM motor over a wide speed range by adaptively adjusting the boundary layer of the switching function. Moreover, the proposed scheme effectively mitigates both steady-state and transient position errors at different speeds by simultaneously considering the phase lag of QSMO and utilizing the proposed NN-based transient position error compensator.

The rest of this paper is organized as follows: In Section 2, the equivalent model of an IPM motor with EEMF and the conventional QSMO-based sensorless drive scheme are introduced. In Section 3, considering the bandwidth and phase lag of the QSMO, a QSMO-based parameter adaptation method and a QSMO-based phase lag compensation method are presented. Considering the PLL model, the design of an NN-based transient position error compensator is presented. In Section 4, the feasibility of the proposed QSMO-based sensorless drive scheme, including parameter adaptation, phase lag compensation, and transient position error compensation methods, is analyzed. In Section 5, the conclusions regarding the proposed sensorless drive scheme are presented.

2. Modeling and Conventional Sensorless Drive Scheme of IPM Motor

2.1. Equivalent Model of IPM Motor with EEMF

In general, the d-q axis model of a three-phase IPM motor can be used for FOC design, thereby enabling operation similar to a DC motor. By using power-variant Park coordinate transformation, the three-phase IPM motor model can be reduced to two dimensions. Then, the voltage equation, electromagnetic torque equation, and mechanical motion equation of the IPM motor in the $dq$ coordinate system can be defined as

$$\left[\begin{array}{c}{u}_{\mathrm{d}}\hfill \\ u_{\mathrm{q}}\hfill \end{array}\right]=\left[\begin{array}{cc}{R}_{\mathrm{s}}+s{L}_{\mathrm{d}}& -L_{\mathrm{q}}{\omega }_{\mathrm{e}}\\ L_{\mathrm{d}}{\omega }_{\mathrm{e}}& R_{\mathrm{s}}+s{L}_{\mathrm{q}}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{d}}\hfill \\ i_{\mathrm{q}}\hfill \end{array}\right]+\left[\begin{array}{c}0\hfill \\ {\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\hfill \end{array}\right],$$

(1)

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}{i}_{\mathrm{q}}\left(i_{\mathrm{d}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)+{\psi }_{\mathrm{f}}\right),$$

(2)

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}=T_{\mathrm{e}}-T_{\mathrm{L}}-B{\omega }_{\mathrm{m}},$$

(3)

and

$${\omega }_{\mathrm{e}}=p_{\mathrm{n}}{\omega }_{\mathrm{m}}$$

(4)

where $u_{\mathrm{d}}$ and $u_{\mathrm{q}}$ are the d- and q-axis stator voltages; $i_{\mathrm{d}}$ and $i_{\mathrm{q}}$ are the d- and q-axis stator currents; $L_{\mathrm{d}}$ and $L_{\mathrm{q}}$ are the d- and q-axis stator inductances; $R_{\mathrm{s}}$ is the stator resistance; ${\psi }_{\mathrm{f}}$ is the permanent magnet flux linkage; $p_{\mathrm{n}}$ is the number of polar pairs; J is the moment of inertia; $T_{\mathrm{e}}$ and $T_{\mathrm{L}}$ are the electromagnetic and load torques; B is the viscous friction coefficient; and ${\omega }_{\mathrm{e}}$ and ${\omega }_{\mathrm{m}}$ are the electrical and mechanical angular speeds.

However, only the $\alpha$-$\beta$ axis model containing the rotor position is suitable for FOC-based sensorless drive design. By using an inverse Park transformation to the d-q axis model in Equation (1), the $\alpha$-$\beta$ axis voltage model of IPM motor can be defined as

$$\left[\begin{array}{c}{u}_{\alpha }\hfill \\ u_{\beta }\hfill \end{array}\right]=R_{\mathrm{s}}\left[\begin{array}{c}{i}_{\alpha }\hfill \\ i_{\beta }\hfill \end{array}\right]+\underset{\mathrm{inductance}}{\underbrace{s{\displaystyle \frac{1}{2}}\left(\left(L_{\mathrm{d}}+L_{\mathrm{q}}\right)+\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)\left[\begin{array}{cc}cos2{\theta }_{\mathrm{e}}& sin2{\theta }_{\mathrm{e}}\\ sin2{\theta }_{\mathrm{e}}& -cos2{\theta }_{\mathrm{e}}\end{array}\right]\right)\left[\begin{array}{c}{i}_{\alpha }\hfill \\ i_{\beta }\hfill \end{array}\right]}}+\underset{\mathrm{EMF}}{\underbrace{{\omega }_{\mathrm{e}}{\phi }_{\mathrm{f}}\left[\begin{array}{c}-sin{\theta }_{\mathrm{e}}\hfill \\ cos{\theta }_{\mathrm{e}}\hfill \end{array}\right]}}$$

(5)

where $u_{\alpha }$ and $u_{\beta }$ are the $\alpha$- and $\beta$-axis stator voltages; $i_{\alpha }$ and $i_{\beta }$ are the $\alpha$- and $\beta$-axis stator currents; and ${\theta }_{\mathrm{e}}$ is the rotor position angle.

Due to the saliency of the IPM motor, both the inductance term and the back EMF term contain the rotor position information. It is difficult to extract the rotor position from two terms simultaneously. To solve this problem, re-constructing the model in Equation (5), the $\alpha$-$\beta$ axis voltage model of the IPM motor with the EEMF can be obtained as

$$\left[\begin{array}{c}{u}_{\alpha }\hfill \\ u_{\beta }\hfill \end{array}\right]=R_{\mathrm{s}}\left[\begin{array}{c}{i}_{\alpha }\hfill \\ i_{\beta }\hfill \end{array}\right]+\left[\begin{array}{cc}s{L}_{\mathrm{d}}& {\omega }_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)\\ -{\omega }_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)& s{L}_{\mathrm{d}}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\hfill \\ i_{\beta }\hfill \end{array}\right]+\underset{\mathrm{EEMF}}{\underbrace{\left(\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)\left({\omega }_{\mathrm{e}}{i}_{\mathrm{d}}-s{i}_{\mathrm{q}}\right)+{\omega }_{\mathrm{e}}{\phi }_{\mathrm{f}}\right)\left[\begin{array}{c}-sin{\theta }_{\mathrm{e}}\hfill \\ cos{\theta }_{\mathrm{e}}\hfill \end{array}\right]}}$$

(6)

where only the EEMF term contains the rotor position information ${\theta }_{\mathrm{e}}$. Furthermore, to simplify the design of QSMO, by neglecting the dynamic characteristics of the d-q axis inductances, the model in Equation (6) can be re-expressed as

$$\left\{\begin{array}{c}{\displaystyle s{i}_{\alpha }=\frac{1}{L_{\mathrm{d}}}\left(u_{\alpha }-R_{\mathrm{s}}{i}_{\alpha }-{\omega }_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)i_{\beta }-e_{\alpha }\right)}\hfill \\ \begin{array}{c}{\displaystyle s{i}_{\beta }=\frac{1}{L_{\mathrm{d}}}\left(u_{\beta }-R_{\mathrm{s}}{i}_{\beta }+{\omega }_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)i_{\alpha }-e_{\beta }\right)}\hfill \\ e_{\alpha }=-\left(\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)\left({\omega }_{\mathrm{e}}{i}_{\mathrm{d}}-s{i}_{\mathrm{q}}\right)+{\omega }_{\mathrm{e}}{\phi }_{\mathrm{f}}\right)sin{\theta }_{\mathrm{e}}\hfill \\ e_{\beta }=\left(\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)\left({\omega }_{\mathrm{e}}{i}_{\mathrm{d}}-s{i}_{\mathrm{q}}\right)+{\omega }_{\mathrm{e}}{\phi }_{\mathrm{f}}\right)cos{\theta }_{\mathrm{e}}\hfill \end{array}\hfill \end{array}\right.$$

(7)

where $e_{\alpha }$ and $e_{\beta }$ are the $\alpha$- and $\beta$-axis EEMF. Finally, the motor model with EEMF in Equation (7) is used to design the QSMO.

2.2. Conventional QSMO-Based Sensorless Drive Scheme

2.2.1. Discrete EEMF-Based QSMO Model

Combined with the motor model with EEMF in Equation (7), correspondingly, the QSMO model for stator current estimation can be obtained as

$$\left\{\begin{array}{c}{\displaystyle s{\widehat{i}}_{\alpha }=\frac{1}{L_{\mathrm{d}}}\left({u_{\alpha }}^{*}-R_{\mathrm{s}}{\widehat{i}}_{\alpha }-{\widehat{\omega }}_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)i_{\beta }-{\widehat{e}}_{\alpha }\right)}\hfill \\ \begin{array}{c}{\displaystyle s{\widehat{i}}_{\beta }=\frac{1}{L_{\mathrm{d}}}\left({u_{\beta }}^{*}-R_{\mathrm{s}}{\widehat{i}}_{\beta }+{\widehat{\omega }}_{\mathrm{e}}\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)i_{\alpha }-{\widehat{e}}_{\beta }\right)}\hfill \\ {\widehat{e}}_{\alpha }=k_{\mathrm{s}}f\left({\tilde{i}}_{\alpha }\right),{\tilde{i}}_{\alpha }={\widehat{i}}_{\alpha }-i_{\alpha }\hfill \\ {\widehat{e}}_{\beta }=k_{\mathrm{s}}f\left({\tilde{i}}_{\beta }\right),{\tilde{i}}_{\beta }={\widehat{i}}_{\beta }-i_{\beta }\hfill \end{array}\hfill \end{array}\right.$$

(8)

where the symbols $\widehat{ }$ and $\widetilde{ }$, respectively, represent the estimated values and the errors between the estimated and actual values of the pertinent variables; $u_{\alpha }^{*}$ and $u_{\beta }^{*}$ are the voltage command outputs from the current closed-loop control; $k_{\mathrm{s}}$ is the sliding mode gain; and $f\left(·\right)$ is the switching function. Compared to other continuous switching functions, the saturation function is selected for the designed QSMO due to its straightforward mathematical expression. The saturation function is defined as

$$f\left({\tilde{i}}_{\alpha \beta }\right)=\mathrm{sat}\left({\tilde{i}}_{\alpha \beta }\right)=\left\{\begin{array}{cc}1,& {\tilde{i}}_{\alpha \beta }>m_{\mathrm{f}}\\ {\displaystyle \frac{{\tilde{i}}_{\alpha \beta }}{m_{\mathrm{f}}},}& \left|{\tilde{i}}_{\alpha \beta }\right|<=m_{\mathrm{f}}\\ -1,& {\tilde{i}}_{\alpha \beta }<-m_{\mathrm{f}}\end{array}\right.$$

(9)

where ${\tilde{i}}_{\alpha \beta }$ is the current error vector and $m_{\mathrm{f}}$ is the boundary layer of the selected switching function. Finally, applying the Euler discrete method to Equations (8) and (9), the EEMF-based QSMO model for discrete system applications, at time step k, can be obtained as

$$\left\{\begin{array}{c}{\displaystyle {\widehat{i}}_{\alpha }\left(k+1\right)=\left({\displaystyle 1-\frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{\mathrm{d}}}}\right){\widehat{i}}_{\alpha }\left(k\right)-\frac{{\widehat{\omega }}_{\mathrm{e}}\left(k\right)\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)T_{\mathrm{s}}}{L_{\mathrm{d}}}{i}_{\beta }\left(k\right)+\frac{T_{\mathrm{s}}}{L_{\mathrm{d}}}\left({u_{\alpha }}^{*}\left(k-1\right)-{\widehat{e}}_{\alpha }\left(k\right)\right)}\hfill \\ {\displaystyle {\widehat{i}}_{\beta }\left(k+1\right)=\left({\displaystyle 1-\frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{\mathrm{d}}}}\right){\widehat{i}}_{\beta }\left(k\right)+\frac{{\widehat{\omega }}_{\mathrm{e}}\left(k\right)\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)T_{\mathrm{s}}}{L_{\mathrm{d}}}{i}_{\alpha }\left(k\right)+\frac{T_{\mathrm{s}}}{L_{\mathrm{d}}}\left({u_{\beta }}^{*}\left(k-1\right)-{\widehat{e}}_{\beta }\left(k\right)\right)}\hfill \\ {\widehat{e}}_{\alpha }\left(k\right)=k_{\mathrm{s}}·\mathrm{sat}\left({\tilde{i}}_{\alpha }\left(k\right)\right)=k_{\mathrm{s}}·\mathrm{sat}\left({\widehat{i}}_{\alpha }\left(k\right)-i_{\alpha }\left(k\right)\right)\hfill \\ {\widehat{e}}_{\beta }\left(k\right)=k_{\mathrm{s}}·\mathrm{sat}\left({\tilde{i}}_{\beta }\left(k\right)\right)=k_{\mathrm{s}}·\mathrm{sat}\left({\widehat{i}}_{\beta }\left(k\right)-i_{\beta }\left(k\right)\right)\hfill \end{array}\right.$$

(10)

where $T_{\mathrm{s}}$ is the discrete period of the controller.

Correspondingly, based on Equation (10), the block diagram of the EEMF-based QSMO at time step k is shown in Figure 1. It is composed of a current estimation unit and an EEMF calculation unit. From Figure 1, the EEMF containing the rotor position information can be obtained in real-time by estimating the stator currents.

2.2.2. QSMO Stability Analysis

Assume that the estimated speed is equal to the actual one and the voltage command outputs are equal to the actual voltages. By subtracting Equation (8) from Equation (7), the error equation of stator currents can be obtained as

$$\left\{\begin{array}{c}{\displaystyle s{\tilde{i}}_{\alpha }=\frac{1}{L_{\mathrm{d}}}\left(-R_{\mathrm{s}}{\tilde{i}}_{\alpha }-{\tilde{e}}_{\alpha }\right)}\hfill \\ {\displaystyle s{\tilde{i}}_{\beta }=\frac{1}{L_{\mathrm{d}}}\left(-R_{\mathrm{s}}{\tilde{i}}_{\beta }-{\tilde{e}}_{\beta }\right)}\hfill \end{array}\right..$$

(11)

The sliding manifold of the QSMO is designed as

$$S=\left[\begin{array}{c}{\tilde{i}}_{\alpha }\hfill \\ {\tilde{i}}_{\beta }\hfill \end{array}\right]=0.$$

(12)

Then, according to the Lyapunov function, two stability conditions of the QSMO can be defined as

$$V={\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}S={\displaystyle \frac{1}{2}}\left({{\tilde{i}}_{\alpha }}^2+{{\tilde{i}}_{\beta }}^2\right)>0$$

(13)

and

$$\begin{array}{c}\dot{V}={\tilde{i}}_{\alpha }\left(s{\tilde{i}}_{\alpha }\right)+{\tilde{i}}_{\beta }\left(s{\tilde{i}}_{\beta }\right)\hfill \\ =-{\tilde{i}}_{\alpha }/\phantom{{\tilde{i}}_{\alpha }{L}_{\mathrm{d}}}{L}_{\mathrm{d}}\left(R_{\mathrm{s}}{\tilde{i}}_{\alpha }+\left({\widehat{e}}_{\alpha }-e_{\alpha }\right)\right)-{\tilde{i}}_{\beta }/\phantom{{\tilde{i}}_{\beta }{L}_{\mathrm{q}}}{L}_{\mathrm{q}}\left(R_{\mathrm{s}}{\tilde{i}}_{\beta }+\left({\widehat{e}}_{\beta }-e_{\beta }\right)\right)\hfill \\ =-R_{\mathrm{s}}/\phantom{R_{\mathrm{s}}{L}_{\mathrm{d}}}{L}_{\mathrm{d}}\left({{\tilde{i}}_{\alpha }}^2+{{\tilde{i}}_{\beta }}^2\right)-1/\phantom{1L_{\mathrm{d}}}{L}_{\mathrm{d}}\left({\widehat{e}}_{\alpha }-e_{\alpha }\right)-1/\phantom{1L_{\mathrm{q}}}{L}_{\mathrm{q}}\left({\widehat{e}}_{\beta }-e_{\beta }\right)\hfill \\ =-\underset{V_1}{\underbrace{R_{\mathrm{s}}/\phantom{R_{\mathrm{s}}{L}_{\mathrm{d}}}{L}_{\mathrm{d}}\left({{\tilde{i}}_{\alpha }}^2+{{\tilde{i}}_{\beta }}^2\right)}}-\underset{V_2}{\underbrace{1/\phantom{1L_{\mathrm{d}}}{L}_{\mathrm{d}}\left(k_{\mathrm{s}}f\left({\tilde{i}}_{\alpha }\right)-e_{\alpha }\right)}}-\underset{V_3}{\underbrace{1/\phantom{1L_{\mathrm{q}}}{L}_{\mathrm{q}}\left(k_{\mathrm{s}}f\left({\tilde{i}}_{\beta }\right)-e_{\beta }\right)}}<0.\hfill \end{array}$$

(14)

Obviously, the condition in Equation (13) is strictly satisfied. In Equation (14), when both the $V_2$ term and the $V_3$ term are positive definite, the sliding mode will be enforced. The system will be converged to the sliding manifold in Equation (12). Hence, the sliding mode gain $k_{\mathrm{s}}$ of the QSMO meets

$$k_s>max(|e_{\alpha }|,|e_{\beta }|).$$

(15)

2.2.3. Conventional Sensorless Drive Scheme

In the conventional scheme, the boundary layer in Equation (9) is determined to be a fixed value ${m_{\mathrm{f}}}^{*}$, through motor operation tests under constant speed [29]. Further, to simultaneously extract speed and position information from the estimated EEMF in Figure 1 and achieve non-delayed position filtering, a PLL is employed. Then, the block diagram of the conventional QSMO-based position and speed estimation method can be obtained as shown in Figure 2. It is composed of a current estimation unit and a position and speed estimation unit. In Figure 2, an LPF is employed to remove high-frequency noise from the estimated speed. The normalization unit is used to constrain the variable amplitude of the estimated EEMF to a constant level. This facilitates the design of PI parameters for the PLL.

Finally, the block diagram of the conventional QSMO-based sensorless drive scheme of IPM motor used for FOC can be obtained as shown in Figure 3. Specifically, the estimated speed information is utilized for speed closed-loop control and decoupling compensation. The estimated position information is employed for coordinate transformation.

2.2.4. Problem Description

In practical applications, the conventional QSMO-based sensorless drive scheme using a PLL, as designed above, exhibits the following problems:

• The fixed boundary layer ${m_{\mathrm{f}}}^{*}$ results in the conventional scheme struggling to operate effectively and stably over a wide range of speeds. Specifically, increases in speed lead to instability in the sensorless drive system.
• The continuous switching function leads to varying steady-state position errors at different constant speeds. It is essential to compensate for steady-state position errors.
• The use of a PLL with low bandwidth introduces noticeable transient position errors during speed variations [30]. To optimize performance, it is essential to compensate for transient position errors.

3. Proposed Adaptive Sensorless Drive with NN-Based Transient Position Error Compensation

3.1. QSMO-Based Parameter Adaptation Method

To achieve the adaptation of the boundary layer $m_{\mathrm{f}}$ and stable operation across a wide range of speeds, the frequency domain model of the QSMO is analyzed in this paper. It avoids reliance on the complex discrete state trajectory analysis [7]. Based on the error equation of stator currents in Equation (11), the transfer function of the QSMO, which is a first-order system, can be obtained as

$$G_{\mathrm{QSMO}}\left(s\right)={\displaystyle \frac{{\widehat{e}}_{\alpha \beta }\left(s\right)}{e_{\alpha \beta }\left(s\right)}}={\displaystyle \frac{k_{\mathrm{s}}/\phantom{k_{\mathrm{s}}{m}_{\mathrm{f}}}{m}_{\mathrm{f}}}{L_{\mathrm{d}}s+\left(k_{\mathrm{s}}/\phantom{k_{\mathrm{s}}{m}_{\mathrm{f}}}{m}_{\mathrm{f}}+R_{\mathrm{s}}\right)}}$$

(16)

where ${\widehat{e}}_{\alpha \beta }$ is the estimated EEMF vector and $e_{\alpha \beta }$ is the actual EEMF vector.

In the conventional scheme, the boundary layer is fixed at ${m_{\mathrm{f}}}^{*}$. The frequency domain analysis results of the EEMF-based QSMO with different speed values ${\omega }_{\mathrm{e}}$ can be obtained as shown in Figure 4a. Combined with the EEMF term in Equation (6) and sliding mode gain $k_{\mathrm{s}}$ in Equation (15), as the speed ${\omega }_{\mathrm{e}}$ increases, the sliding mode gain $k_{\mathrm{s}}$ increases along with the EEMF. The increase in the sliding mode gain $k_{\mathrm{s}}$ results in a rightward shift of the magnitude-frequency characteristic and an increase in the QSMO bandwidth ${\omega }_{\mathrm{QSMO}}$. However, increasing the bandwidth heightens sensitivity to high-frequency noise and reduces the robustness of the estimation system. Specifically, this results in a reduced filtering capability of the QSMO for the input EEMF. To address this drawback, the optimal QSMO bandwidth value prior to the speed change should be preserved. It will ensure that the estimation system maintains consistent robust performance across different speeds. Therefore, this paper selects a pertinent fixed bandwidth ${{\omega }_{\mathrm{QSMO}}}^{*}$, instead of the fixed boundary layer ${m_{\mathrm{f}}}^{*}$. Based on the transfer function in Equation (16), the bandwidth of the QSMO can be obtained as

$${\omega }_{\mathrm{QSMO}}={\displaystyle \frac{k_{\mathrm{s}}/\phantom{k_{\mathrm{s}}{m}_{\mathrm{f}}}{m}_{\mathrm{f}}+R_{\mathrm{s}}}{L_{\mathrm{d}}}}.$$

(17)

Finally, based on Equation (17) and under the fixed bandwidth ${{\omega }_{\mathrm{QSMO}}}^{*}$, the proposed QSMO-based parameter adaptation method for operation over a wide range of speeds can be defined as

$$m_{\mathrm{f}}={\displaystyle \frac{k_{\mathrm{s}}}{L_{\mathrm{d}}{{\omega }_{\mathrm{QSMO}}}^{*}-R_{\mathrm{s}}}}.$$

(18)

where the boundary layer $m_{\mathrm{f}}$ can be dynamically and adaptively adjusted, instead of using the fixed value ${m_{\mathrm{f}}}^{*}$, when the sliding mode gain $k_{\mathrm{s}}$ changes with variations in speed.

3.2. QSMO-Based Phase Lag Compensation Method

From Figure 4b, when the bandwidth is fixed, the frequency domain characteristics of the QSMO will be uniquely determined. Consequently, the phase lag ${\theta }_{\mathrm{QSMO}}$ will increase with the speed ${\omega }_{\mathrm{e}}$. Hence, to compensate for steady-state position errors over a wide range of speeds, the phase lag of the QSMO under the fixed bandwidth ${{\omega }_{\mathrm{QSMO}}}^{*}$ should be considered as

$${\theta }_{\mathrm{QSMO}}\left({\omega }_{\mathrm{e}}\right)=-arctan\left({\displaystyle \frac{{\omega }_{\mathrm{e}}}{{{\omega }_{\mathrm{QSMO}}}^{*}}}\right)$$

(19)

where the speed ${\omega }_{\mathrm{e}}$ is also the frequency of the EEMF.

However, the actual speed ${\omega }_{\mathrm{e}}$ cannot be obtained in the sensorless drive system. When the speed is stably and accurately estimated, the estimated speed ${\widehat{\omega }}_{\mathrm{e}}$ can be used to design the phase lag compensation. Finally, based on Equation (19), the proposed QSMO-based phase lag compensation method can be defined as

$${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}1}={\theta }_{\mathrm{QSMO}}\left({\widehat{\omega }}_{\mathrm{e}}\right)=-arctan\left({\displaystyle \frac{{\widehat{\omega }}_{\mathrm{e}}}{{{\omega }_{\mathrm{QSMO}}}^{*}}}\right).$$

(20)

This equation can be used to compensate for varying steady-state position errors at different constant speeds.

3.3. NN-Based Transient Position Error Compensation Method

3.3.1. Architecture

To compensate for transient position errors and avoid the complexity of frequency domain analysis, this paper proposes a transient position error compensator based on a feedback time delay neural network (FB-TDNN). Compared to a standard back propagation neural network (BP-NN) without delay taps and feedback, the FB-TDNN offers superior capabilities in long-term dependency learning and time series data modeling. Before designing the compensator, it is essential to select a variable related to transient position errors as the input signal for the FB-TDNN.

Obviously, transient position errors are caused by speed variations during acceleration and deceleration. Thus, the PLL related to speed extraction in Figure 2 should be analyzed. The specific block diagram of the PLL with a speed LPF is shown in Figure 5.

In Figure 5, $G_{\mathrm{LPF}\_{\widehat{\omega }}_{\mathrm{e}}}\left(s\right)$ is the transfer function of the speed LPF; $K_{\mathrm{p}\_\mathrm{PLL}}$ and $K_{\mathrm{i}\_\mathrm{PLL}}$ are the PI parameters of the PLL; ${\widehat{e}}_{\beta }^n$ and ${\widehat{e}}_{\alpha }^n$ are the normalized EEMF; ${\theta }_{\mathrm{e}}$ is the rotor position from the estimated EEMF before extraction; ${\widehat{\omega }}_{\mathrm{e}\_\mathrm{PLL}}$ and ${\widehat{\theta }}_{\mathrm{e}}$ are the extracted speed and position from the PLL; ${\widehat{\omega }}_{\mathrm{e}}$ is the final estimated speed obtained after simple low-pass filtering; and $\Delta e$ is the normalized EEMF error.

According to the PLL model with a speed LPF in Figure 5, the equivalent transfer function of the PLL can be obtained as

$$G_{\mathrm{PLL}}\left(s\right)={\displaystyle \frac{{\widehat{\theta }}_{\mathrm{e}}\left(s\right)}{{\theta }_{\mathrm{e}}\left(s\right)}}={\displaystyle \frac{K_{\mathrm{p}\_\mathrm{PLL}}s+K_{\mathrm{i}\_\mathrm{PLL}}}{s^2+K_{\mathrm{p}\_\mathrm{PLL}}s+K_{\mathrm{i}\_\mathrm{PLL}}}}$$

(21)

where ${\widehat{\theta }}_{\mathrm{e}}$ satisfies ${\widehat{\omega }}_{\mathrm{e}}/\phantom{{\widehat{\omega }}_{\mathrm{e}}{\widehat{\theta }}_{\mathrm{e}}}{\widehat{\theta }}_{\mathrm{e}}=s{G}_{\mathrm{LPF}\_{\widehat{\omega }}_{\mathrm{e}}}$; ${\theta }_{\mathrm{e}}$ satisfies $\Delta e=sin\left({\theta }_{\mathrm{e}}-{\widehat{\theta }}_{\mathrm{e}}\right)\approx {\theta }_{\mathrm{e}}-{\widehat{\theta }}_{\mathrm{e}}$, when ${\theta }_{\mathrm{e}}-{\widehat{\theta }}_{\mathrm{e}}$ approaches zero; the position error is defined as ${\theta }_{\mathrm{e}\_\mathrm{err}}={\widehat{\theta }}_{\mathrm{e}}-{\theta }_{\mathrm{e}}$. Based on Equation (21), the transfer function from the estimated acceleration ${\widehat{a}}_{\mathrm{e}}$ to the position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$ can also be obtained as

$${\displaystyle \frac{{\widehat{a}}_{\mathrm{e}}\left(s\right)}{-{\theta }_{\mathrm{e}\_\mathrm{err}}\left(s\right)}}={\displaystyle \frac{s{\widehat{\omega }}_{\mathrm{e}}\left(s\right)}{\Delta e\left(s\right)}}=G_{\mathrm{LPF}\_{\widehat{\omega }}_{\mathrm{e}}}\left(s\right)\left(s{K}_{\mathrm{p}\_\mathrm{PLL}}+K_{\mathrm{i}\_\mathrm{PLL}}\right)$$

(22)

where ${\widehat{a}}_{\mathrm{e}}$ satisfies ${\widehat{a}}_{\mathrm{e}}\left(s\right)=\Delta {\widehat{\omega }}_{\mathrm{e}}\left(s\right)/\phantom{\Delta {\widehat{\omega }}_{\mathrm{e}}\left(s\right)\Delta t}\Delta t$, and the time difference $\Delta t$ is the discrete period $T_{\mathrm{s}}$. Then, based on Equation (22), the transfer function from the position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$ to the speed difference $\Delta {\widehat{\omega }}_{\mathrm{e}}$ can be obtained as

$${\displaystyle \frac{-{\theta }_{\mathrm{e}\_\mathrm{err}}\left(s\right)}{\Delta {\widehat{\omega }}_{\mathrm{e}}\left(s\right)}}={\displaystyle \frac{1}{G_{\mathrm{LPF}\_{\widehat{\omega }}_{\mathrm{e}}}\left(s\right)·T_{\mathrm{s}}·\left(s{K}_{\mathrm{p}\_\mathrm{PLL}}+K_{\mathrm{i}\_\mathrm{PLL}}\right)}}.$$

(23)

Obviously, from Equation (23), there is a positive correlation between the position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$ and the speed difference $-\Delta {\widehat{\omega }}_{\mathrm{e}}$, described as

$${\theta }_{\mathrm{e}\_\mathrm{err}}={\widehat{\theta }}_{\mathrm{e}}-{\theta }_{\mathrm{e}}\propto -\Delta {\widehat{\omega }}_{\mathrm{e}}.$$

(24)

Finally, based on Equation (24), the speed difference $-\Delta {\widehat{\omega }}_{\mathrm{e}}$ is selected as the input for the proposed NN-based transient position error compensator. The block diagram of the proposed compensator is shown in Figure 6. It is composed of a feedforward neural network which is a standard BP-NN, an input tapped delay line (TDL), and a feedback TDL.

In Figure 6, at time step k, the input signal is the selected speed difference $-\Delta {\widehat{\omega }}_{\mathrm{e}}\left(k\right)$ between the estimated speeds ${\widehat{\omega }}_{\mathrm{e}}(k-1)$ and ${\widehat{\omega }}_{\mathrm{e}}\left(k\right)$ of the two states. The output signal is the estimated transient position error compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right)$. The input TDL includes m time series input signals, such as $-\Delta {\widehat{\omega }}_{\mathrm{e}}\left(k\right)$, $-\Delta {\widehat{\omega }}_{\mathrm{e}}(k-1)$, …, and $-\Delta {\widehat{\omega }}_{\mathrm{e}}(k-m+1)$. The feedback TDL includes n time series feedback signals derived from the estimated compensation output, such as ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-1)$, ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-2)$, …, and ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-n)$. It can be used to enhance the capability of capturing long-term dependencies. The loss function at time step k is the mean squared error (MSE) between the actual transient position error ${\theta }_{\mathrm{e}\_\mathrm{err}}\left(k\right)$ and the estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right)$. It can be used to guide offline training of the proposed FB-TDNN.

Moreover, the structure of the proposed FB-TDNN is shown in Figure 7. It consists of one input layer, two hidden layers, and one output layer. The input layer contains $m+n$ signals from the input TDL and feedback TDL. The number of hidden layers is two, due to its stronger approximation ability than a one-hidden-layer neural network [35]. The number of neurons in each hidden layer is determined to be ten by gradually increasing the number of neurons and evaluating them step-by-step. The activation function of each hidden layer is selected as the Rectified Linear Unit (ReLU) function. The straightforward mathematical expression of the ReLU function makes it particularly suitable for discrete algorithm design. The neuron of the output layer has no activation function and no bias. The offline training method of the proposed FB-TDNN is the back propagation (BP) learning algorithm.

3.3.2. Discrete BP Learning Algorithm for Offline Training

The discrete BP learning algorithm for offline training can be obtained based on the proposed FB-TDNN. In detail, the discrete algorithm includes three steps in each iteration, namely, forward propagation, backward propagation, and gradient descent. Forward propagation is used to output the estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right)$ with updated parameters; back propagation and gradient descent are specific steps to update the parameters of the FB-TDNN, based on the loss function.

The forward propagation of the i-th layer at time step k, for obtaining ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right)$, is given as

$$\begin{array}{cc}\left\{\begin{array}{c}{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)·{\mathbf{Y}}_{\left(n\left[i-1\right],1\right)}^{\left[i-1\right]}\left(k\right)+{\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)\hfill \\ {\mathbf{Y}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\sigma }^{\left[i\right]}\left({\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)\right)\hfill \end{array}\right.,& i=1,2\end{array}$$

(25)

and

$$\begin{array}{cc}{\mathbf{Y}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)·{\mathbf{Y}}_{\left(n\left[i-1\right],1\right)}^{\left[i-1\right]}\left(k\right),& i=3\end{array}$$

(26)

where i indicates the i-th layer; $n\left[i\right]$ indicates the size of the i-th layer; the numbers inside parentheses $\left(n\left[i\right],n\left[i-1\right]\right)$ indicate the dimension of a vector or matrix; and ${\mathbf{W}}_{\left(n\left[i+1\right],n\left[i\right]\right)}^{\left[i+1\right]}\left(k\right)$, ${\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)$, ${\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)$, ${\sigma }^{\left[i\right]}\left(·\right)$ and ${\mathbf{Y}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)$ are the weight, bias, activation function’s input, activation function and output of the i-th layer, respectively. The input of the first layer and the output of the third layer at time step k are defined as

$$\begin{array}{c}{\mathbf{Y}}_{\left(m+n,1\right)}^{\left[0\right]}\left(k\right)=\begin{array}{cccc}[-\Delta {\widehat{\omega }}_{\mathrm{e}}\left(k\right)& -\Delta {\widehat{\omega }}_{\mathrm{e}}(k-1)& ...& -\Delta {\widehat{\omega }}_{\mathrm{e}}(k-m+1)\end{array}\\ \begin{array}{cccc}{\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-1)& {\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-2)& ...& {\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(k-n){]}^{\mathrm{T}}\end{array}\end{array}$$

(27)

and

$${\mathbf{Y}}_{\left(1,1\right)}^{\left[3\right]}\left(k\right)={\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right).$$

(28)

The back propagation of the i-th layer at time step k, for calculating the gradient, is given as

$$\begin{array}{cc}\left\{\begin{array}{c}d{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)=d{\mathbf{Y}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(k\right)-{\theta }_{\mathrm{e}\_\mathrm{err}}\left(k\right)\hfill \\ d{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)=d{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)·{\left({\mathbf{Y}}_{\left(n\left[i-1\right],1\right)}^{\left[i-1\right]}\left(k\right)\right)}^{\mathrm{T}}\hfill \end{array}\right.,& i=3\end{array}$$

(29)

and

$$\begin{array}{cc}\left\{\begin{array}{c}d{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)={\left({\mathbf{W}}_{\left(n\left[i+1\right],n\left[i\right]\right)}^{\left[i+1\right]}\left(k\right)\right)}^{\mathrm{T}}·d{\mathbf{Z}}_{\left(n\left[i+1\right],1\right)}^{\left[i+1\right]}\left(k\right)\odot {{\sigma }^{\prime }}^{\left[i\right]}\left({\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)\right)\hfill \\ d{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)=d{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)·{\left({\mathbf{Y}}_{\left(n\left[i-1\right],1\right)}^{\left[i-1\right]}\left(k\right)\right)}^{\mathrm{T}}\hfill \\ d{\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)=d{\mathbf{Z}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)\hfill \end{array}\right.,& i=1,2\end{array}$$

(30)

where the symbol ⊙ is a Hadamard product.

The gradient descent of the i-th layer at time step k, used to update weights and biases for the next time step, is given as

$$\begin{array}{cc}{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}(k+1)={\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)-\alpha ·d{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right),& i=3\end{array}$$

(31)

and

$$\begin{array}{cc}\left\{\begin{array}{c}{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}(k+1)={\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)-\alpha ·d{\mathbf{W}}_{\left(n\left[i\right],n\left[i-1\right]\right)}^{\left[i\right]}\left(k\right)\hfill \\ {\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}(k+1)={\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)-\beta ·d{\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(k\right)\hfill \end{array}\right.,& i=1,2\end{array}$$

(32)

where the hyperparameters $\alpha$ and $\beta$ determine the step size of updating weights and biases.

In the initial state, the parameters ${\mathbf{B}}_{\left(n\left[i\right],1\right)}^{\left[i\right]}\left(1\right)$ and ${\mathbf{W}}_{\left(n\left[i+1\right],n\left[i\right]\right)}^{\left[i+1\right]}\left(1\right)$ need to be initialized randomly. The variables that cannot be obtained in the initial state, such as ${\widehat{\omega }}_{\mathrm{e}}\left(0\right)$, ${\widehat{\omega }}_{\mathrm{e}}(-1)$, ${\widehat{\omega }}_{\mathrm{e}}(-2)$, …, ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\left(0\right)$, ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(-1)$, ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}(-2)$, …, are all set to 0. After offline training is completed, Equations (25)–(28), along with the well-trained parameters, constitute the discrete algorithm of the proposed NN-based transient position error compensator.

3.3.3. Offline Training Process

The specific structure of the proposed FB-TDNN, which are numbers of the input delay taps (IDTs) and feedback delay taps (FDTs), and the optimal parameters, should be determined during the offline training process.

Firstly, preliminary preparations for training the FB-TDNN are necessary. An offline training scenario with fully transient characteristics is selected, where the rotational speed varies between 1500 rpm and 2000 rpm every 0.1 s. It is noted that the employed scenario is obtained through the implementation of the proposed sensorless drive scheme. This scheme utilizes QSMO-based parameter adaptation and phase lag compensation methods. The estimated rotational speed ${\widehat{N}}_{\mathrm{r}}$, position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$, and speed difference $-\Delta {\widehat{\omega }}_{\mathrm{e}}$ in the selected offline training scenario are shown in Figure 8. In Figure 8, the input data which are the speed difference $-\Delta {\widehat{\omega }}_{\mathrm{e}}$ and the training target, which is the actual position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$, are both selected as the dataset. The dataset is divided into training and validation sets in a typical ratio of 3:1. The hyperparameters $\alpha$ and $\beta$ are determined to be fixed values without acceleration optimization. Two performance evaluation metrics, which are the MSE and generalization gap, are selected to assess the proposed FB-TDNN. The MSE used to evaluate the performance and the generalization gap G used to evaluate the generalization ability are defined as

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left({\theta }_{\mathrm{e}\_\mathrm{err}}-{\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}\right)}^2$$

(33)

and

$$G=\left({\displaystyle \frac{{\mathrm{MSE}}_{\mathrm{val}}-{\mathrm{MSE}}_{\mathrm{train}}}{{\mathrm{MSE}}_{\mathrm{train}}}}\right)\times 100\%$$

(34)

where n is the number of data points used to compute the MSE; ${\mathrm{MSE}}_{\mathrm{train}}$ is the MSE for the training set; and ${\mathrm{MSE}}_{\mathrm{val}}$ is the MSE for the validation set.

Secondly, combined with the algorithm in Equations (25)–(32) and the two metrics in Equations (33) and (34), the numbers of the IDTs and FDTs for the proposed FB-TDNN will be determined by a trial-and-error method in a MATLAB script. Specifically, to determine the optimal number of IDTs, the FB-TDNN is tested by gradually increasing the number of IDTs from 1 until the bound of short-term dependencies is found. In this case, the FB-TDNN without FDTs is similar to a BP-NN. Once the number of IDTs is determined, the optimal number of FDTs is then found by testing the FB-TDNN. This involves gradually increasing the number of FDTs from 0 until the performance of the FB-TDNN shows no significant improvement. Finally, the training results with different IDTs and FDTs can be obtained as shown in Figure 9 and Figure 10. These results include the MSE-epoch curves during offline training and estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}$. The evaluation results of the FB-TDNN with different IDTs and FDTs can also be obtained as shown in

Table 1. Evaluation results of the FB-TDNN with different IDTs and FDTs.

Metrics	1 IDT	2 IDTs	3 IDTs	4 IDTs 1	3 IDTs	3 IDTs	3 IDTs
	0 FDTs	0 FDTs	0 FDTs	0 FDTs	1 FDT	2 FDTs	3 FDTs
${\mathrm{MSE}}_{\mathrm{train}}$	$1.592328\times {10}^{-5}$	$9.179970\times {10}^{-6}$	$5.091869\times {10}^{-6}$	None	$4.778487\times {10}^{-6}$	$3.119331\times {10}^{-6}$	$1.753465\times {10}^{-6}$
${\mathrm{MSE}}_{\mathrm{val}}$	$1.808068\times {10}^{-5}$	$9.984575\times {10}^{-6}$	$5.520496\times {10}^{-6}$	None	$5.097090\times {10}^{-6}$	$3.187434\times {10}^{-6}$	$1.799001\times {10}^{-6}$
G	13.6%	8.8%	8.4%	None	6.7%	2.2%	2.6%

¹ FB-TDNN with 4 IDTs and 0 FDTs exhibits no learning capability in the selected training scenario.

.

From Figure 10a–c, despite the absence of FDTs, disturbances in the estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}$ noticeably decrease as IDTs increase from 1 to 3. From Figure 9a–c and Table 1, ${\mathrm{MSE}}_{\mathrm{train}}$ decreases from $1.592328\times {10}^{-5}$ to $5.091869\times {10}^{-6}$, and ${\mathrm{MSE}}_{\mathrm{val}}$ decreases from $1.808068\times {10}^{-5}$ to $5.520496\times {10}^{-6}$ as the number of IDTs increases. Therefore, the training accuracy of the FB-TDNN can be enhanced by increasing the number of IDTs. However, when the number of IDTs reaches 4 (over 3), FB-TDNN exhibits no learning capability whatsoever in the selected training scenario. Hence, the number of IDTs is finally determined to be 3.

From Figure 10c,d, when the number of IDTs is 3, the addition of an FDT significantly reduces disturbances in the estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}$. From Figure 10d,f, disturbances in the compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}$ further decrease as FDTs increase from 1 to 3. From Figure 9c–f, the training speed of the FB-TDNN are improved as the number of FDTs increases from 0 to 3. When going from no FDTs to an FDT, the training speed improves most significantly. From Table 1, ${\mathrm{MSE}}_{\mathrm{train}}$ decreases from $5.091869\times {10}^{-6}$ to $1.753465\times {10}^{-6}$, and ${\mathrm{MSE}}_{\mathrm{val}}$ decreases from $5.520496\times {10}^{-6}$ to $1.799001\times {10}^{-6}$ as the number of FDTs increases from 0 to 3. However, from the generalization gap G, when the number of FDTs reaches 3, the generalization ability has no significant improvement. Thus, the number of FDTs is finally determined to be 3.

Thirdly, after the specific structure is determined, it is crucial to determine the optimal model parameters. This ensures that the proposed NN-based transient position error compensator exhibits good generalization capability. In detail, the FB-TDNN should be retrained in the selected scenario. Then, at the end of each epoch, updated model parameters are used to evaluate the FB-TDNN. This evaluation is performed on both the training and validation sets, yielding the ${\mathrm{MSE}}_{\mathrm{train}}$ and ${\mathrm{MSE}}_{\mathrm{val}}$. Finally, the MSE-epoch curves including the training MSE, training set MSE (${\mathrm{MSE}}_{\mathrm{train}}$), and the validation set MSE (${\mathrm{MSE}}_{\mathrm{val}}$) can be obtained as shown in Figure 11a. Different from the MSE-epoch curve for the training set MSE (orange), the curve for the training MSE (blue) is derived by employing model parameters updated in real-time within each epoch, similar to Figure 9.

To prevent over-fitting, the optimal model parameters are typically determined based on the epoch with the minimum validation set MSE, marked in Figure 11a. After integrating the selected parameters into the FB-TDNN, the estimated compensation ${\widehat{\theta }}_{\mathrm{e}\_\mathrm{comp}2}$ and the position error after offline compensation can be obtained as shown in Figure 11b. In Figure 11b, ${\mathrm{MSE}}_{\mathrm{train}}$ and ${\mathrm{MSE}}_{\mathrm{val}}$ are $1.796264\times {10}^{-6}$ and $1.792371\times {10}^{-6}$, respectively. The generalization gap G is −0.2%, which is below 0. Therefore, theoretically, the proposed NN-based transient position error compensator has robust generalization ability under the selected parameters.

4. Simulation and Analysis

According to the conventional scheme in Figure 3 and the proposed three methods in Section 3.3, a platform for feasibility validation is built in Matlab Simulink, as shown in Figure 12. The built platform utilizes a low-frequency c-code controller (5 kHz) and a high-frequency control plant (10MHz) to simulate embedded control. Specifically, it is composed of a c-code controller, an SVM modulator, an IPM motor, and a three-phase inverter. The c-code controller is used to achieve speed and current control, as well as position and speed estimation.

The main parameters in the simulation are listed in

Table 2. Main parameters in the simulation.

Parameters	Value
Rated load torque, $T_{\mathrm{d}}$	5 $\mathrm{N}$$·$$\mathrm{m}$
Rated speed range, $N_{\mathrm{r}}$ 1	1500–2000 rpm
Frequency of controller and PWM, $f_{\mathrm{s}}$	5 kHz
Rated dc voltage, $U_{\mathrm{dc}}$	311 V
Nominal d-axis stator inductance, $L_{\mathrm{d}}$	1.20 mH
Nominal d-axis stator inductance, $L_{\mathrm{q}}$	2.00 mH
Nominal flux linkage, ${\psi }_{\mathrm{f}}$	0.052 Wb
Nominal stator resistance, $R_{\mathrm{s}}$	0.343 $\Omega$
Number of polar pairs, $p_{\mathrm{n}}$	4
Solver step size	${10}^{-7}$ s

¹ It is a standard speed range for a commercial electrical equipment application, such as an air compressor. The design margin, which is the maximum speed range, is reserved for 2000–2250 rpm.

. In Table 2, in order to verify the proposed scheme, the rated speed range and maximum speed range of the air compressor from a cooperating company are taken as practical conditions. The rated speed range is 1500–2000 rpm, and the maximum speed range is 2000–2250 rpm. The feasibility validation of the proposed scheme will be conducted by sequentially enabling the proposed three methods, step-by-step.

4.1. Feasibility Analysis of the QSMO-Based Parameter Adaptation Method

To validate the feasibility of the proposed parameter adaptation method, a wide-speed operation scenario should be considered, where the speed rises from the lower boundary to the upper boundary of the rated speed range, as shown in Table 2. Specifically, in this scenario, the rotational speed command suddenly rises from 1500 rpm to 2000 rpm at 1 s. The test results of the proposed QSMO-based parameter adaptation method are shown in Figure 13. These results include the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, actual speed $N_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$, boundary layer $m_{\mathrm{f}}$, sliding mode gain $k_{\mathrm{s}}$, estimated EEMF ${\widehat{e}}_{\alpha }$ and ${\widehat{e}}_{\beta }$, and QSMO bandwidth ${\omega }_{\mathrm{QSMO}}$. It is noted that the sliding mode gain $k_{\mathrm{s}}$ suitable for practical discrete applications in [7] is adopted, along with a ramp-up and ramp-down approach to envelop the EEMF, as shown in Figure 13b.

From Figure 13a, before the speed rise, the sensorless drive system operates normally with the well-designed fixed boundary layer ${m_{\mathrm{f}}}^{*}$. Specifically, the torque $T_{\mathrm{e}}$ can be stably maintained at 5 $\mathrm{N}$$·$$\mathrm{m}$, and the speed $N_{\mathrm{r}}$ can be kept at 1500 rpm. There is no obvious speed error $N_{\mathrm{r}\_\mathrm{err}}$, and the position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$ remains constant. In Figure 13b, the well-designed fixed boundary layer ${m_{\mathrm{f}}}^{*}$ ensures that the sliding mode gain $k_{\mathrm{s}}$ wraps the EEMF well, allowing for stable estimation of the EEMF. However, after the speed rises to 2000 rpm, the QSMO bandwidth ${\omega }_{\mathrm{QSMO}}$ increases with the sliding mode gain $k_{\mathrm{s}}$. This leads to a decrease in the noise suppression capability of the QSMO, resulting in chattering in the EEMF. As a result, there are oscillations in the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, and position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$, as shown in Figure 13a. At 1.25 s, the QSMO-based parameter adaptation method is enabled. From Figure 13b, the QSMO bandwidth ${\omega }_{\mathrm{QSMO}}$ drops to the value before the speed rise. The chattering in the EEMF is also mitigated. From Figure 13a, the sensorless drive system can operate as stably as before, even with the speed $N_{\mathrm{r}}$ now kept at 2000 rpm. In detail, the oscillations in the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, and position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$ disappeared. Therefore, the proposed QSMO-based parameter adaptation method enables stable wide-speed operation of the IPM motor.

4.2. Feasibility Analysis of the QSMO-Based Phase Lag Compensation Method

To validate the feasibility of the proposed phase lag compensation method, both the lower and upper boundaries of the rated speed range in Table 2 should be evaluated. Specifically, this involves testing the method in two steady-state scenarios with rotational speed commands set to 1500 and 2000 rpm, respectively. Test results of the proposed QSMO-based phase lag compensation method are shown in Figure 14. These results include the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, actual speed $N_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, and position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$.

In these two scenarios, the proposed QSMO-based parameter adaptation method is enabled throughout. Therefore, in Figure 14, the torque $T_{\mathrm{e}}$ can be stably held at 5 $\mathrm{N}$$·$$\mathrm{m}$ throughout. In Figure 14a and Figure 14b, the speed values are also stably kept at 1500 and 2000 rpm, respectively. The steady-state speed errors are both 0. However, comparing Figure 14a and Figure 14b, different steady-state speeds result in different steady-state position errors: 0.08 rad and 0.10 rad, respectively. These errors cannot be compensated with a constant value. From 0.75 s to 1.25 s, the QSMO-based phase lag compensation method is enabled for both scenarios. From Figure 14a and Figure 14b, under the proposed method, the steady-state position errors decrease from 0.08 rad and 0.1 rad to 0 rad, respectively. Moreover, the proposed method does not affect the speed estimation of the conventional sensorless scheme, i.e., the speed error $N_{\mathrm{r}\_\mathrm{err}}$ is still 0. Therefore, the proposed adaptive scheme, incorporating the QSMO-based parameter adaptation and phase lag compensation methods, achieves sensorless drive over a wide speed range without steady-state speed and position errors.

4.3. Feasibility Analysis of the NN-Based Transient Position Error Compensation Method

Furthermore, the proposed adaptive scheme, which uses the QSMO-based parameter adaptation and phase lag compensation methods, is tested in another scenario. In this scenario, the rotational speed command varies between 1500 rpm and 2000 rpm every 0.3 s. The test results can be obtained as shown in Figure 15a. These results include the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, actual speed $N_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, and position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$. From Figure 15a, thus far, the proposed adaptive scheme demonstrates favorable steady-state characteristics for torque $T_{\mathrm{e}}$ and speed $N_{\mathrm{r}}$. The steady-state speed and position errors remain at 0, and the torque $T_{\mathrm{e}}$ is stably maintained at 5 $\mathrm{N}·\mathrm{m}$ across different speed values. However, transient position errors persist due to speed variations. To mitigate transient position errors, the proposed NN-based transient position error compensator is enabled. Test results similar to Figure 15a can be obtained as shown in Figure 15b. Additionally, the estimated EEMF ${\widehat{e}}_{\alpha }$ and ${\widehat{e}}_{\beta }$ after transient position error compensation can also be obtained as shown in Figure 15c.

In Figure 15b, although the speed variation interval increases from 0.1 s in the training scenario (Figure 8) to 0.3 s in this scenario (Figure 15b), the average transient error time is reduced by at least 90%. The average maximum transient error is also reduced by almost 77%. In addition, the proposed compensator does not affect the speed estimation. The steady-state speed and position errors are still 0. Moreover, in Figure 15c, the EEMF can also be stably estimated and clearly exhibits good sinusoidal characteristics without oscillations.Therefore, the proposed NN-based transient error compensator can effectively mitigate transient position errors without introducing additional static errors and oscillations.

4.4. Feasibility Analysis under a Simulated Wide-Speed Application Scenario

To further effectively validate the proposed scheme and ensure scientific rigor, a simulated wide-speed application scenario is generated randomly and objectively. These speed values in this scenario are random but fall within the standard speed range (1500–2000 rpm) and the maximum speed range (2000–2250 rpm) of the air compressor. Specifically, in this scenario, the speed command suddenly varies from different values every 0.3 s: 1700 rpm–2250 rpm–1600 rpm–1950 rpm–1500 rpm. Then, the test results before and after transient position error compensation can be obtained as shown in Figure 16a and Figure 16b, respectively. These results include the torque $T_{\mathrm{e}}$, estimated speed ${\widehat{N}}_{\mathrm{r}}$, actual speed $N_{\mathrm{r}}$, speed error $N_{\mathrm{r}\_\mathrm{err}}$, and position error ${\theta }_{\mathrm{e}\_\mathrm{err}}$. The estimated EEMF ${\widehat{e}}_{\alpha }$ and ${\widehat{e}}_{\beta }$ after the compensation can also be obtained as shown in Figure 16c.

In Figure 16a, the proposed adaptive scheme, including the QSMO-based parameter adaptation and phase lag compensation methods, continues to demonstrate favorable and stable performance. In detail, despite random speed variations, the IPM motor operates stably without steady-state speed and position errors, and the torque $T_{\mathrm{e}}$ is stably kept at 5 $\mathrm{N}$$·$$\mathrm{m}$. The torque $T_{\mathrm{e}}$ and speed $N_{\mathrm{r}}$ still exhibit good steady-state characteristics. In Figure 16b, the proposed NN-based transient position error compensation method is enabled. Comparing speed errors in both Figure 16a,b, after enabling transient position error compensation, the sensorless drive system still maintains good performance in speed estimation. Although the speed command values and speed variations in this scenario differ from those in the training scenario (Figure 8), the average transient error time is also reduced by at least 90%. The average maximum transient error is reduced by 56%. Therefore, the proposed NN-based compensator can be considered an effective method for mitigating transient position errors and has strong generalization ability. Additionally, in Figure 16c, the excellent sinusoidal characteristics at the maximum speed (2250 rpm) indicate that the EEMF can be stably estimated under the proposed scheme, without any chattering.

In conclusion, in the simulated wide-speed application scenario for the air compressor, the proposed scheme exhibits good stability and effectively mitigates rotor position errors across a wide range of speeds. Therefore, the proposed scheme is feasible and can be used in commercial electrical equipment.

5. Conclusions

This paper proposed an adaptive QSMO-based sensorless drive scheme of an IPM motor with NN-based transient position error compensation for FOC. The analysis of QSMO magnitude-frequency characteristics enables the proposed adaptation method to effectively mitigate instability caused by speed (bandwidth) increases. The analysis of QSMO phase-frequency characteristics enables the proposed phase lag compensation method to effectively mitigate varying steady-state position errors at different speeds. The analysis of the PLL confirms the possibility of designing an NN-based transient position error compensator. Combined with the trial-and-error method and offline BP learning algorithm, the −0.2% generalization gap indicates that the proposed compensator exhibits robust generalization ability and enables transient position error mitigation.

In simulated experiments involving sudden changes in speed, the results confirmed that the proposed QSMO-based parameter adaptation and phase lag compensation methods enable stable sensorless drive over a wide speed range without steady-state position errors; the proposed NN-based transient position error compensation method can effectively mitigate transient position errors. Finally, in a simulated wide-speed application scenario for the air compressor, the results confirmed that the proposed scheme demonstrates robust stability and effectively mitigates rotor position errors across a wide range of speeds. Future work will involve further optimizing the proposed algorithm for implementation in the actual embedded controller, conducting experiments, and applying it to the air compressor from our cooperating company.