# Online Inductance Identification of Permanent Magnet Synchronous Motor Based on Extended Kalman Filter With Dead-Time Compensation

Kai Yang

School of Electrical and
Electronic Engineering,
Huazhong University of
Science and Technology

WuHan, China
yk@hust.edu.cn

Guoyi Qiu

School of Electrical and
Electronic Engineering,
Huazhong University of
Science and Technology

WuHan, China
qiuguoyi@hust.edu.cn

Cheng Luo

School of Electrical and
Electronic Engineering,
Huazhong University of
Science and Technology

WuHan, China
luoxcheng@hust.edu.cn

## Keywords

«Permanent magnet motor», «Parameter identification», «Dead-time», «Kalman Filter».

## Abstract

A dq-axis inductance identification method is proposed for permanent magnet synchronous motor (PMSM), which is based on an extended Kalman filter with dead-time compensation. The proposed method is intended to estimate the disturbance voltage in the dq coordinate system, which is caused by dead-time and non-ideal switching characteristics of the power devices. The proposed method is achieved by means of parametric calculation, which is then fed into the Kalman filter to compensate the dead-time effect. Furthermore, the sensitivity analyses of the input parameters demonstrate that the accuracy of the extended Kalman filter discrimination is contingent on the precision of the input parameters. The effectiveness of the proposed method is demonstrated by simulation results.

## Introduction

Due to high torque density and high efficiency, PMSM have been widely used in various occasions. The design of the parameters of PMSM controller often depends on the specific parameters of the motor, and the changes in the motor parameters due to temperature, load and other factors will have a certain impact on the performance of the motor controller. It is pointed out in [1] that parameter changes will lead to a decrease in the efficiency and current regulation

performance of PMSM. Accurate PMSM parameters are often the key to ensure system stability, efficiency and dynamic response in [2]. Therefore, it is necessary to identify the relevant parameters of PMSM online to ensure the normal operation of PMSM.

The on-line parameter identification method of PMSM should estimate the motor parameters in real time without affecting the operation of the motor. The numerical method is one of the more commonly used methods. The numerical methods are recursive least squares (RLS), extended Kalman filter (EKF) and model reference adaptive system (MRAS). Firstly, RLS tries to find a model that minimizes the sum of squares of the residuals between the predicted values and the actual observed values of the model [3]. In the field of motor control, RLS is often used to estimate PMSM parameters to achieve sensorless control algorithms [4]. Secondly, EKF is a special least squares estimator based on recursive algorithm, which mainly contains two steps: prediction and update, due to the problem of motor equation underrank, two Kalman filters are used in [5] to identify multiple motor parameters at the same time. In addition to being used for motor parameter identification, EKF is also used to identify mechanical parameters, such as moment of inertia, viscosity coefficient [6]. Finally, MRAS adjusts the controller parameters by comparing the difference between the actual system output and the model reference output, and uses Lyapunov's stability theorem to design the system to reduce the error to close to 0 to read the relevant parameter values of the parameter estimator [7]. In [8], cascaded MRAS was used to simultaneously estimate the rotor speed, stator resistance, and rotor flux. The performance of

<footnote>This work was supported in part by National Key Research and Development Program of China (2023YFB4301500); in part by National Natural Science Foundation of China (52207055).</footnote>
MRAS and EKF is compared and discussed in [9].

The inverter output voltage consists of high-frequency discrete pulses that are difficult to measure, so the researchers used the reference voltage as the discriminator input. Actually, discrepancies between the reference and actual voltages arise from factors such as dead time, IGBT turn-off time, and voltage drops, necessitating compensation. The effects of dead time and power device nonlinearities on inverter output voltage are thoroughly analyzed in [10]. In [11], an interference observer is employed online to estimate interference voltages in the rotor reference frame caused by dead time and non-ideal switching characteristics of power devices, which are then fed into the voltage reference to compensate for dead time effects.

In this thesis, based on the mathematical model of PMSM, the EKF algorithm model as well as the effects of dead-time and nonlinear characteristics of the power devices on $U_d$, $U_q$ are analyzed. The corresponding compensating measures are proposed, which are finally applied to the identification model of the dq-axis inductance of the PMSM. The results of the compensated EKF discriminator are compared with those of the uncompensated EKF discriminator, and the simulation proves the effectiveness of the proposed method.

## Mathematical model of PMSM

The PMSM current mathematical model in the dq reference frame can be expressed as follows:

$$ \frac{di_d}{dt} = \frac{1}{L_d} u_d - \frac{R}{L_d} i_d + \frac{L_q}{L_d} w_r i_q \quad (1) $$

$$ \frac{di_q}{dt} = \frac{1}{L_q} u_q - \frac{R}{L_q} i_q - \frac{L_d}{L_q} w_r i_d - \frac{\Psi_f w_r}{L_q} \quad (2) $$

where $i_d, i_q, u_d$ and $u_q$ are currents and voltages in dq axis respectively, $L_d, L_q$ are inductances in dq axis respectively, $R$ is the stator resistance, $\Psi_f$ is the permanent magnet flux linkage, $\omega_r$ is the synchronous electrical angular velocity.

## VSI Nonlinearities

The effect of dead-time and voltage drop on the inverter output voltage is analysed by taking the inverter A-phase bridge arm as an example. During the dead time, two switches of the same arm are turned off, and one of the diodes is turned on. If the current flows to the load, the

lower arm diode is turned on, otherwise, the upper arm diode is turned on. Fig. 1 shows the basic configuration of the inverter single-phase bridge arm, defining the positive direction of the A-phase current as the flow from the power supply to the load.

<figure><img src="image_1.png" /><figcaption>Fig. 1: Basic configuration of one phase leg of PWM inverter.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2: Neglect the turn on/off times of switching devices. (a) Ideal gate signal patterns. (b) Practical gate signal pattern considering the dead time. (d) Ideal output voltage for $i_A > 0$. (d) Actual output voltage considering the dead time for $i_A > 0$.</figcaption></figure>

Fig. 2(a) shows the ideal gate signal pattern and Fig. 2(b) shows the practical gate signal pattern considering the dead time, and Fig. 2(c) and (d) show the output voltages in ideal condition and in condition considering dead-time for $i_A > 0$ respectively. Moreover, when the phase current is positive/negative, the phase current flows through the bottom/top diode during the dead-time period.

When the phase current $i_A$ is positive, the turn-on time error $T_{A,error}$ and the phase A voltage error $V_{A,error}$ can be calculated as follows:

$$ T_{A,error} = T_A^+ - T_{A,dead}^+ = T_{dead} \quad (3) $$
$$V_{A,error}^{+} = V_{As} - V_{As,dead} = \frac{T_{A}^{+} - T_{A,dead}^{+}}{T_{s}} V_{dc} \quad (4)$$

Similarly, when the phase current $i_A$ is negative, the turn-on time error $T_{A,error}$ and the phase A voltage error $V_{A,error}$ can be calculated as follows:

$$T_{A,error}^{-} = T_{A}^{-} - T_{A,dead}^{-} = T_{dead} \quad (5)$$

$$V_{A,error}^{-} = V_{As} - V_{As,dead} = \frac{T_{A}^{-} - T_{A,dead}^{-}}{T_{s}} V_{dc} \quad (6)$$

Equations (3)~(6) can be summarized as

$$T_{A,error} = T_{dead} \cdot \operatorname{sgn}(i_A) \quad (7)$$

$$V_{A,error} = \frac{T_{A,error}}{T_s} \cdot V_{dc} = V_{dead} \cdot \operatorname{sgn}(i_A) \quad (8)$$

$$\operatorname{sgn}(i_A) = \begin{cases} 1 & : i_A > 0 \\ -1 & : i_A < 0 \end{cases}$$

where

$$V_{dead} = \frac{T_{dead}}{T_s} \cdot V_{dc}$$

Additionally, the voltage drops of switching device can be considered as follows:

$$V_{dead} = \frac{T_{dead}}{T_s} \cdot (V_{dc} - V_{sat} + V_d) + \frac{V_{sat} + V_d}{2} \quad (9)$$

where $V_{sat}$ and $V_d$ are a saturation voltage drop of the active switch and a forward voltage drop of the freewheeling diode, respectively.

Distortion voltages in the dq reference frame can be expressed as follows:

$$\begin{pmatrix} u_{d,dead} \\ u_{q,dead} \end{pmatrix} = V_{dead} P(\theta_e) [\operatorname{sgn}(i_A), \operatorname{sgn}(i_B), \operatorname{sgn}(i_C)]^T \quad (10)$$

where $P(\theta_e)$ represents the Park transformation.

## EKF model

The EKF is a set of real-time recursive algorithms, and in a nonlinear discrete system, it contains two basic expressions, which can be expressed as follows:

(1) State Equation:

$$x_k = f(x_{k-1}, u_{k-1}, w_{k-1}) \quad (11)$$

(2) Observation Equation:

$$y_k = h(x_k, v_k) \quad (12)$$

where $x_k$ is the state quantity at moment k, $y_k$ is the observation at moment k, $x_{k-1}$ is the state quantity at moment k-1, $u_{k-1}$ is the input at moment k-1, and $w_{k-1}, v_k$ are zero-mean white

noise sequences with covariance matrices Q and R respectively.

In order to linearize the nonlinear system, the nonlinear functions f and h Taylor are expanded, and the corresponding Jacobian matrices are obtained by ignoring the higher order terms, and (11) and (12) can be reduced to (13) and (14).

(1) State Equation:

$$x_k = A(k-1)x_{k-1} + B(k-1)u_{k-1} + w_{k-1} \quad (13)$$

(2) Observation Equation:

$$y_k = H(k-1)x_k + v_k \quad (14)$$

$$A(k-1) = \frac{\partial f(x_{k-1}, u_{k-1}, w_{k-1})}{\partial x_{k-1}}$$

Where $B(k-1) = \frac{\partial f(x_{k-1}, u_{k-1}, w_{k-1})}{\partial u_k}$

$$H(k) = \frac{\partial h(x_k, v_k)}{\partial x_k}$$

The calculation process of the EKF is a continuous "prediction-update" process, and the equations for the EKF prediction loop for a nonlinear discrete system are as follows:

$$x(k/k-1) = f(x(k-1), u(k-1)) \quad (15)$$

$$P(k/k-1) = AP(k-1)A^T + Q \quad (16)$$

The updated link equations are as follows:

$$K(k) = \frac{P(k/k-1)H^T}{HP(k/k-1)H^T + R} \quad (17)$$

$$x(k) = x(k/k-1) + K(k)[z(k) - Hx(k/k-1)] \quad (18)$$

$$P(k) = [I - K(k)H]P(k/k-1) \quad (19)$$

where $k/k-1$ is a prior estimate of k-moment filtering to predict k-time, $k-1$ is the state of the relevant physical quantity at the time of k-1, $k$ is the state of the relevant physical quantity at time k, $P$ is the filter error covariance matrix, $K$ is the Kalman filter gain, $Q$ is the process noise covariance matrix, $R$ is the Measurement noise covariance matrix. The flow of EKF algorithm is shown in Fig. 3.

The flow of EKF algorithm is shown in Fig. 3.

<figure><img src="image_3.png" /><figcaption>Fig.3: Extended Kalman Filter Flow Diagram.</figcaption></figure>
Identification Model of dq axis
Inductance

When identifying the parameters of the salient-
pole PMSM, the parameters that need to be
identified are the stator resistance R, the rotor
flux Ψ<sub>f</sub>, the d-axis inductance L<sub>d</sub>, and the q-axis
inductance L<sub>q</sub>. Because there are only two
equations of state of the salient-pole PMSM, it is
impossible to identify multiple parameters at the
same time. The resistance and flux generally do
not change much during the actual motor
operation, the values of those are set to constant
in the parameter identification process, and only
the dq axis inductance are identified.

Since the identification step size is usually much
smaller than the electrical parameters of the
motor, it is considered that the electrical
parameters do not change during the
identification process, and their differential
values are approximately 0.

The d-axis current i<sub>d</sub>, q-axis current i<sub>q</sub>, d-axis inductance L<sub>d</sub>, and q-axis inductance L<sub>q</sub> of the permanent magnet synchronous motor are taken as state variables. The d-axis current i<sub>d</sub> and q-axis current i<sub>q</sub> are used as observational measurements. The identification model of the q-axis inductance L<sub>q</sub> and the d-axis inductance L<sub>d</sub> are established under the dq axis, as follows:

$$
\left\{
\begin{aligned}
\frac{di_d}{dt} &= \frac{u_d - u_{d,dead}}{L_d} - \frac{R}{L_d}i_d + \frac{L_q}{L_d}w_r i_q \\
\frac{di_q}{dt} &= -\frac{R}{L_q}i_q - \frac{L_d}{L_q}w_r i_d - \frac{\Psi_f}{L_q}w_r + \frac{u_q - u_{q,dead}}{L_q} \\
\frac{dL_d}{dt} &= 0 \\
\frac{dL_q}{dt} &= 0
\end{aligned}
\right.
\qquad (20)
$$

Discretization of (20) yields the following
equation:

$$
\left\{
\begin{aligned}
i_d(k) &= \frac{T_s(u_d(k-1) - u_{d,dead}(k-1))}{L_d(k-1)} \\
&\quad + \left(1 - \frac{RT_s}{L_d(k-1)}\right)i_d(k-1) + \frac{L_q(k-1)T_s}{L_d(k-1)}w_r i_q(k-1) \\
i_q(k) &= \left(1 - \frac{RT_s}{L_q(k-1)}\right)i_q(k-1) - \frac{L_d(k-1)T_s}{L_q(k-1)}w_r i_d(k-1) \\
&\quad - \frac{\Psi_f T_s}{L_q(k-1)}w_r + \frac{T_s(u_q(k-1) - u_{q,dead}(k-1))}{L_q(k-1)} \\
L_d(k) &= L_d(k-1) \\
L_q(k) &= L_q(k-1)
\end{aligned}
\right.
\qquad (21)
$$

where $T_s$ is the sampling time.

The selected state variable is x=[i<sub>d</sub> i<sub>q</sub> L<sub>d</sub> L<sub>q</sub>]<sup>T</sup>, the
input variable u=[u<sub>d</sub> u<sub>q</sub>]<sup>T</sup>, and the observed
variable is y=[i<sub>d</sub> i<sub>q</sub>]<sup>T</sup>.

Solving the Jacobian matrix of the system, it
obtains the linear system matrix A and the
output matrix H :

$$
\begin{align*}
A = \frac{\partial f}{\partial x} &=
\begin{bmatrix}
1 - \frac{RT_s}{L_d(k-1)} & \frac{L_q(k-1)T_s w_r}{L_d(k-1)} & F_1 & F_2 \\
-\frac{L_d(k-1)T_s w_r}{L_q(k-1)} & 1 - \frac{RT_s}{L_q(k-1)} & F_3 & F_4 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} \\
F_1 &= \frac{-T_s(u_d(k-1) - u_{d,dead}(k-1)) + RT_s i_d(k-1)}{L_d^2(k-1)} \tag{22} \\
&\quad - \frac{L_q(k-1)T_s w_r i_q(k-1)}{L_d^2(k-1)} \\
F_2 &= \frac{T_s w_r i_q(k-1)}{L_d(k-1)}, \quad F_3 = \frac{-T_s w_r i_d(k-1)}{L_q(k-1)} \\
F_4 &= \frac{RT_s i_q(k-1) + L_d(k-1)T_s w_r i_d(k-1)}{L_q^2(k-1)} \\
&\quad + \frac{\Psi_f T_s w_r - T_s(u_q(k-1) - u_{q,dead}(k-1))}{L_q^2(k-1)} \\
H &= \frac{\partial h}{\partial x} =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{bmatrix}
\tag{23}
\end{align*}
$$

<figure><img src="image_4.png" /><figcaption>Fig.4: Block diagram of motor control with EKF identifier containing dead-time compensation.</figcaption></figure>
## Simulation result

In order to verify the effectiveness of the identification model based on EKF proposed in this paper, a PMSM vector control system is established as shown in Fig.4, which includes modules such as PI control loop, SVPWM driver, dead-time compensator and EKF parameter identifier. The reference output voltages after the current loop PI controller are fed into the EKF recogniser for operation after passing through the dead-time compensator.

The parameters to be input in the EKF model are the initial value of the state variable $x_0$, the initial value of the covariance matrix $P_0$, the process error matrix $Q$ and the measurement error matrix $R$. Table I shows the values of the motor and switching devices parameters used in this paper, and the $x_0$, $P_0$, $Q$ and $R$ of the EKF identifier is shown in Table II. The DC bus voltage of VSI is 150V, and the frequency is 10kHz. Performance of the identifier is tested by several scenarios, namely, the constant speed test, the speed step test, and the load step test.

<table><caption>TABLE I. Motor and switching devices parameters</caption><thead><tr><td>Name</td><td>Parameter value</td></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>0.9664Ω</td></tr><tr><td>Flux</td><td>0.09382Wb</td></tr><tr><td>L<sub>d</sub></td><td>4.24mH</td></tr><tr><td>L<sub>q</sub></td><td>6.21mH</td></tr><tr><td>Dead-time</td><td>1.5μs</td></tr><tr><td>V<sub>sat</sub></td><td>1.2V</td></tr><tr><td>V<sub>d</sub></td><td>1.2 V</td></tr></tbody></table>

<table><caption>TABLE II. EKF parameters</caption><thead><tr><td>Name</td><td>value</td></tr></thead><tbody><tr><td>x<sub>0</sub></td><td>[0,0,0,0]<sup>T</sup></td></tr><tr><td>P<sub>0</sub></td><td>diag[1,1,1,1]</td></tr><tr><td>Q</td><td>diag[1e-8,1e-8,0.1,0.1]</td></tr><tr><td>R</td><td>diag[1e-6,1e-6]</td></tr></tbody></table>

### A. Constant Speed & Constant Load

The first test is conducted with the PMSM running at a constant speed. A load torque of 2N.m is applied to the motor speed at 750rpm as a given speed. The EKF identifier starts working in 0.1s.

<figure><img src="image_5.png" /><figcaption>Fig.5: Identified parameters of the PMSM with constant speed. Offline measured value in black. The EKF without dead-time compensation in brown. The EKF with dead-time compensation in purple and denoted with EKFC. (a) Estimated d-axis inductance $L_d$. (b) Estimated q-axis inductance $L_q$.</figcaption></figure>

The identification results are shown in Fig.5. In Fig.5(a), the EKF identifier without dead-time compensation with the estimated d-axis inductance value 18.54mH has a magnificent offset at the steady state from the nominal value 4.24mH. The dual EKF identifier with dead-time compensation(4.38mH) is much more accurate. In Fig.5(b), in the same case, EKFC is more accurate than EKF, and the EKF identifier with dead-time compensation is more stability and less fluctuation in identification result.

### B. Speed Step

The PMSM is given a load is set to 2N.m and a speed step from 750 to 1000 r/min at time t=0.3s. The identification results are shown in Fig.6.

<figure><img src="image_6.png" /></figure>
Fig.6: Identified parameters of the PMSM with step speed. (a) Estimated d-axis inductance L<sub>d</sub>. (b) Estimated q-axis inductance L<sub>q</sub>.

As shown in Fig.6, the EKF without dead-time compensation converges after speed step within 0.6s, while the EKFC reaches new steady state within 0.46s. Besides, compared with the EKF without dead-time compensation, EKFC has less overshoot during the transient, indicating that the EKFC has better dynamic performance.

### C. Load Step

The motor speed is set at 750rpm. The PMSM is given a load step from 2 to 4 N.m at time t=0.3s. The identification results are shown in Fig.7.

<figure><img src="image_7.png" /><figcaption>Fig.7: Identified parameters of the PMSM with step load. (a) Estimated d-axis inductance L<sub>d</sub>. (b) Estimated q-axis inductance L<sub>q</sub>.</figcaption></figure>

As shown in Fig.7, the EKF without dead-time compensation reaches new steady state after load step within 0.26s, while the EKFC has little change after load step. Compared with speed step, the EKF without dead-time compensation and the EKFC have less overshoot during the transient.

### D. Sensitivity to Input Values

The Kalman filter-based parameter estimator relies on the estimation results of the previous steps in performing the motor model identification, and therefore, in order to ensure the reliability and accuracy of the estimation process, it is necessary to explore in detail the influence of the accuracy of the input quantities on the output results of the EKF. In the actual operating environment of the motor, the resistance value and the permanent flux linkage will dynamically fluctuate with the changes in

the operating conditions of the motor, thus potentially affecting the accuracy of the inductor discrimination. Therefore, the focus of this section is to analyze the specific effects of resistance and magnetic chain variations on inductor discrimination.

In this section, it is assumed that the motor operates at a constant rotational speed (750 rpm) with a constant load (2 Nm) to simulate its actual operating conditions. To further explore the effects of these variations on the inductance discrimination, Fig. 8 demonstrates the trend of the inductance discrimination results during the variation of the resistance value from 90% to 110%, while Fig. 9 shows the response characteristics of the inductance discrimination results under the condition of permanent flux linkage variation from 90% to 110%.

<figure><img src="image_8.png" /><figcaption>Fig.8: Identified parameters of the PMSM with variable resistance. The EKFC with 90% resistance in purple. The EKFC with 100% resistance in brown. The EKFC with 110% resistance in blue. (a) Estimated d-axis inductance L<sub>d</sub>. (b) Estimated q-axis inductance L<sub>q</sub>.</figcaption></figure>

<figure><img src="image_9.png" /></figure>
Fig.9: Identified parameters of the PMSM with variable flux. (a) Estimated d-axis inductance L<sub>d</sub>. (b) Estimated q-axis inductance L<sub>q</sub>.

As showed in Fig.8 and Fig.9, in EKFC, the small variations of the input resistance and permanent flux linkage have a significant effect on the identification results of the d-axis inductor, but have almost no effect on the identification results of the q-axis inductor. This phenomenon indicates that the identification process of the d-axis inductance is more sensitive to the input resistance and permanent flux linkage, while the q-axis inductance shows more stability. More specifically, the same degree of variation of the permanent flux linkage has a more significant effect on the inductor discrimination in the EKFC compared to the variation of the resistance. For example, in Fig. 8a, the identification result of the d-axis inductance is 5.864 mH when the resistance is varied up to 90%, which is significantly closer to the reference value of the inductance (4.24 mH), whereas in Fig. 9 (a), when the permanent flux linkage is varied up to 90%, the identification result of the d-axis inductance is 13.98 mH, which showing a large deviation. This shows that the effect of permanent flux linkage variation on the inductance discrimination is much larger than the resistance variation.

Conclusion

In order to obtain the inductance value of the dq-
axis of a PMSM in real time, this paper
establishes an EKF model with dead-time
compensation for identifying the inductance of
the dq-axis and compares it with the EKF
identifier without dead-time compensation, and
the simulation results show the effectiveness of
the proposed dead-time compensation method.
In addition, this paper analyzes and compares
the effects produced by the variation of
resistance and permanent flux linkage in the
input parameters, and proves the disadvantage of
the EKF algorithm that the output accuracy is
seriously dependent on the input accuracy.

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