Sliding Mode Control for Sensorless Speed Tracking of PMSM with Whale Optimization Algorithm and Extended Kalman Filter

Abstract

This paper proposes a sensorless speed control strategy for a permanent magnet synchronous motor system. Sliding mode control with a whale optimization algorithm was developed for robustness and chattering reduction. To estimate the position and speed of the rotor, an extended Kalman filter using Gaussian process regression was designed. In this controller, the whale optimization method adjusts the switching gain to minimize the tracking error. However, it provides chattering reduction and robustness, owing to the adaptive gain. The extended Kalman estimator calculates the rotor speed by using the current and voltage of the motor as an observer. The observer ensures the high reliability and low cost of the controller. The noise covariance and weight matrices that validated the performance of the estimation were optimized using a regression algorithm. The Gaussian process regression was trained to approximate the best covariance and matrices from the results of the motor controller execution. The performance of the proposed method was demonstrated through simulations under several conditions of tracking speed and load torque changes.

1. Introduction

Permanent magnet synchronous motors (PMSMs) are used in various industrial fields because of their high efficiency, precision, and power density [1]. However, the PMSM speed regulation system includes parameter uncertainties, perturbations, and external load disturbances that degrade the control performance [2]. Therefore, it is difficult to control the PMSM with high precision using a linear control method [3]. Recently, several nonlinear control strategies have been proposed to improve the control performance of PMSMs. These include passivity control [4], neural network-based control [5], predictive control [6], and fuzzy control [7].

Among these methods, sliding mode control (SMC) has been considered as a solution with the advantages of quick response, simplicity, and high accuracy [8]. The SMC method is well known for its disturbance rejection and robustness [9]. However, the chattering phenomenon of the SMC originating from the switching gain is the main issue that limits its performance [10]. The large switching gain ensures the robustness of the SMC and can excite high-frequency dynamics [11]. Several methods have been developed to reduce chattering [12]. In [13], a high-order SMC was presented to eliminate the discontinuous term in the control input. In [14], backstepping control was designed to reduce chattering. The proposed filtered SMC, which provides a reconstructed signal, is an outstanding method [15]. For the sliding variable design, a disturbance estimator is applied to the sliding mode controller [16]. This study proposes an adaptive switching gain with super-twisting reaching law-based SMC using a whale optimization algorithm (WOA) [17]. The WOA is a metaheuristic algorithm that mimics the hunting behavior of humpback whales to obtain an optimal solution. Many metaheuristic algorithms have been applied for solving nonlinear and multimodal optimization problems due to their simple and flexible characteristics. Such algorithms including Crow Search Algorithm, Galactic Swarm Optimization, and Butterfly Optimization Algorithm are used in various industrial fields [18]. Compared to these algorithms, WOA is easier to implement and more flexible. Moreover, the WOA is gradient-free and insensitive to the initial solutions [19]. With the advantage of a quick and effective response, the WOA is used to optimize the switching gain of the SMC according to the speed error. However, these controllers require information about the rotor speed and position. Position sensors reduce the reliability of a system and increase costs [20]. The malfunction of the sensors may limit the PMSM controller’s operation. Therefore, the reduction of sensor dependence is necessary for reliability enhancement [21]. Therefore, sensorless control is important in PMSMs. Sensorless control algorithms include model reference adaptive control [22], back electromotive force [23], a state observer such as the adaptive interconnected observer [24], and a sliding mode observer [25]. The sensorless control methods have been developed to be suitable in wide-ranging speeds and robust to load variations.

The extended Kalman filter (EKF) is an optimal estimator for sensorless controllers. The EKF provides the rotor speed and angle from the stator current and voltage, and is robust and effective [26]. The EKF updates the estimation of the rotor state using the measured information and zero-mean multivariate Gaussian noise covariance. The advantage of EKF is that it can support estimation even when the input and model are not precise, which makes the algorithm popular [27]. However, to achieve sufficient performance, tuning of the EKF covariance is needed [28]. Gaussian process regression (GPR) was proposed to optimize the parameters of the EKF. The GPR is a function approximator that uses a training dataset and a Gaussian distribution model [29]. GPR is applied to predict the state of many systems with the advantages that the parameters can be adaptively obtained. Moreover, GPR’s output prediction is related to probability distribution [30]. Using prior knowledge, GPR generates several Gaussian distributions of the dataset and predicts the uncertain position of the new data. GPR has been applied to improve the control performance in industrial fields [31].

In this study, WOA-SMC and GPR-EKF were combined to improve the speed regulation performance of a sensorless PMSM control. A WOA that optimizes the switching gain was developed to reduce chattering, response time, and control effort. A GPR that tuned the noise covariance matrices of the EKF was applied to provide accurate rotor speed results for the PMSM controller. Using these methods, the proposed controller improves the performance with accurate speed estimation, robustness, reduced chattering, and quick response time. The main contributions that this paper proposes can be written as follows:

• The chattering from SMC is reduced by using WOA, which can adjust switching gain with error. The optimization based on WOA can provide adaptive gains using system parameters.
• The performance of PMSM sensorless control using EKF is improved by applying the GPR, which can optimize noise matrices. The GPR selects the proper noise matrices to maximize EKF accuracy.
• The optimized gains and noise matrices can enhance the sensorless control for PMSM. The speed tracking performance with WOA-SMC and GPR-EKF shows the fast speed attainment and robustness against external load and speed changes.

The remainder of this paper is organized as follows: In Section 2, the PMSM model and problem formulation are presented. The super-twisting reaching law (STRL)-based SMC and WOA are presented in Section 3. Section 4 describes the EKF algorithm, GPR, and how the GPR-EKF cooperates. The simulation results are presented in Section 5. Finally, the conclusions are presented in Section 6.

2. System Description

2.1. Permanent Magnet Synchronous Motor Modeling

The mathematical model of PMSM in d-q coordinates can be expressed as follows:

$$\left\{\begin{array}{c}{u}_d=R_s{i}_d-\omega L_q{i}_q+L_d\frac{d{i}_d}{dt},\hfill \\ u_q=R_s{i}_q+\omega L_d{i}_d+\omega \psi +L_q\frac{d{i}_q}{dt},\hfill \end{array}\right.$$

(1)

where $u_d$ and $u_q$ are the stator voltages of the d-q axes, respectively, and $R_s$ is the stator resistance. $i_d$ and $i_q$ represent d- and q-axes currents, respectively. $\omega$ is the mechanical angular speed of the rotor, and $\psi$ denotes the flux of the rotor. $L_d$ and $L_q$ are the stator inductances of the d- and q-axes, respectively. For the surface-mounted PMSM, the d- and q-axes inductances are equal ($L_d=L_q$). Then, the electromagnetic torque $T_e$ can be represented as follows:

$$T_e=\frac{3}{2}p\psi i_q,$$

(2)

where p denotes the number of pole pairs. From Equation (1), the motion dynamics can be expressed as

$$\frac{d\omega }{dt}=\frac{T_e}{J}-\frac{B}{J}\omega -\frac{T_L}{J},$$

(3)

where J denotes rotational inertia, B is the friction coefficient, and $T_L$ means the load torque.

2.2. Sensorless Controller Design for Permanent Magnet Synchronous Motor

Rotor information is required for sensorless control. The EKF can estimate the rotor speed using the PMSM voltage and current. The GPR tunes the EKF covariance matrices to achieve an effective performance. The WOA-SMC regulates the reference current at the rotor speed, which is the output of the GPR-EKF. The overall block diagram of the proposed method for sensorless PMSM control is shown in Figure 1, where ${\omega }_{ref}$ is the reference speed and $\widehat{\omega }$ is the estimated speed.

3. Sliding Mode Control Design with Whale Optimization Algorithm

3.1. Super-Twisting Reaching Law-Based Sliding Mode Control

The main goal of the proposed controller is the speed tracking and regulation of the PMSM. The speed error is defined as follows:

$$e={\omega }_{ref}-\omega ,$$

(4)

The sliding surface is designed as

$$s=\dot{e}+c_{1}e+c_2{e}^{\frac{p}{q}},$$

(5)

where $c_1>0$, $c_2>0$, p and q denote positive odd numbers. The super-twisting reaching law is defined as follows:

$$\dot{s}=-k_1{\left|s\right|}^{\frac{1}{2}}sgn\left(s\right)-k_2{\int }_0^{t}sgn\left(s\right)d\tau ,$$

(6)

where $k_1$ and $k_2$ are positive switching gain values. Then, the output of the controller is derived as follows:

$$i_q^{*}=\frac{2J}{3p\psi }{\int }_0^t\left(c_1\dot{e}+c_2\frac{d{e}^{\frac{p}{q}}}{dt}+k_1{\left|s\right|}^{\frac{1}{2}}sgn\left(s\right)+k_2\int sgn\left(s\right)d{\tau }_1\right)d{\tau }_2.$$

(7)

3.2. Whale Optimization Algorithm

The WOA is a meta-heuristic optimization algorithm [32]. Figure 2 illustrates the procedure of WOA.

The WOA can be divided into three phases. The first phase is the encircling prey phase, which is represented by the following equations:

$$\begin{array}{c}D=\left|C·X^{*}\left(t\right)-X\left(t\right)\right|,\hfill \\ X(t+1)=X^{*}-\left(A·D\right),\hfill \end{array}$$

(8)

where D is the distance between the target and the optimized position of this iteration, t indicates the current iteration, C and A represent the coefficients, and X is the best position that has been optimized so far. C and A are calculated as follows:

$$\begin{array}{c}C=2r,\hfill \\ A=2a·r-a,\hfill \end{array}$$

(9)

where a is a variable that linearly decreases from 2 to 0 and r is randomly chosen.

The second phase is the bubble-net attack method. The shrinking encircling and spiral updating position mechanisms were used in the exploitation phase. In shrinking encircling, the value of A decreases over the iterations, and the new position of the search agent appears anywhere between the current and optimal positions. The distance between the current and optimal positions and movement were calculated in the spiral updating position phase. The mathematical model is described as follows:

$$X(t+1)=\left\{\begin{array}{c}{X}^{*}\left(t\right)-A·D,\hfill \\ D^{\prime }·e^{bl}·cos\left(2\pi l\right)+X^{*}\left(t\right),\hfill \end{array}\right.\begin{array}{c}ifp<0.5,\\ ifp\ge 0.5,\end{array}$$

(10)

where b is a constant, l is a random number within $[-1,1]$, and p is a random number within $[0,1]$. When $\left|A\right|>1$, the position of the search agent in the bubble-net attack phase is updated based on a randomly chosen search agent. The mathematical model of the prey search phase, which is the third phase, can be described as follows:

$$\begin{array}{c}D=\left|C·X_{rand}\left(t\right)-X\left(t\right)\right|,\hfill \\ X\left(t+1\right)=X_{rand}\left(t\right)-A·D,\hfill \end{array}$$

(11)

where $X_{rand}$ is a randomly selected position vector that can be used instead of the best optimized value so far. If $X_{rand}$ is a better solution than the previous X, the parameter is updated. The WOA updates all the parameters for each iteration. In the proposed method, the algorithm determines the fitness and position of the optimal values. We set the PMSM controller fitness function to minimize the controller output and error. The fitness function is expressed as follows:

$${\int }_0^t{\left(\alpha {\left(i_q^{*}-i_q\right)}^2+{\left({\omega }_{ref}-\widehat{\omega }\right)}^2\right)}^{\frac{1}{2}}d\tau ,$$

(12)

where $\alpha$ denotes a positive constant. The WOA then optimizes the switching gains $k_1$ and $k_2$ that reduce the rotor speed error. During the iterations, the WOA determines the best solution for the controller. Figure 3 shows the scheme for the WOA-SMC.

3.3. Stability Analysis

To verify the stability of the proposed controller, Equation (6) can be rewritten as [33]:

$$\left\{\begin{array}{c}{\dot{x}}_1=-k_1{\left|s\right|}^{\frac{1}{2}}sgn\left(s\right)+x_2,\hfill \\ {\dot{x}}_2=-k_{2}sgn\left(s\right).\hfill \end{array}\right.$$

(13)

We set the vector $\zeta$ as follows:

$${\zeta }^T=\left[\begin{array}{cc}{\zeta }_1& {\zeta }_2\end{array}\right],$$

(14)

where ${\zeta }_1=|x_1{|}^{1/2}sgn\left(x_1\right)$ and ${\zeta }_2=x_2$. The derivate of Equation (14) is given as follows:

$$\begin{array}{cc}\dot{\zeta }=\frac{1}{\left|{\zeta }_1\right|}\chi \zeta ,& \chi =\left[\begin{array}{cc}-\frac{1}{2}{k}_1& \frac{1}{2}\\ -k_2& 0\end{array}\right]\end{array}.$$

(15)

Then Lyapunov function is selected as follows:

$$V={\zeta }^T\eta \zeta ,$$

(16)

where $\eta$ is a constant symmetric positive definite matrix. Then, it is shown that the proposed controller is stable, as follows:

$$\dot{V}=-{\left|x_1\right|}^{\frac{1}{2}}{\zeta }^T\lambda \zeta ,$$

(17)

where ${\chi }^T\eta +\eta \chi =-\lambda$. $\chi$ is Hurwitz if $k_1$ and $k_2$ are positive. We can obtain $\dot{V}$ as negative with the condition of $\chi$.

4. Extended Kalman Filter Design with Gaussian Process Regression

4.1. Extended Kalman Filter

The nonlinear dynamic model of PMSM can be written as follows [34]:

$$\begin{array}{c}\dot{\mathbf{x}}\left(t\right)=f\left(\mathbf{x}\left(t\right),\mathbf{u}\left(t\right)\right)+\mathbf{w}\left(t\right),\hfill \\ {\mathbf{z}}_k=h\left({\mathbf{x}}_k\right)+{\mathbf{v}}_k,\hfill \end{array}$$

(18)

where $\mathbf{w}$ and $\mathbf{v}$ denote random disturbances. $\mathbf{w}$ and $\mathbf{v}$ are the process and measurement noise, respectively. The choice of covariance matrices is critical for EKF performance. The noise covariance can be written as follows:

$$\begin{array}{c}\mathbf{w}\left(t\right)\sim N\left(0,\mathbf{Q}\left(t\right)\right),\hfill \\ {\mathbf{v}}_k\sim N\left(0,{\mathbf{R}}_k\right).\hfill \end{array}$$

(19)

Then, we can set the system states as follows:

$$\begin{array}{c}\mathbf{x}={\left[i_d,i_q,\omega ,\theta \right]}^T,\hfill \\ \mathbf{u}={\left[u_d,u_q\right]}^T,\hfill \\ \mathbf{y}={\left[i_d,i_q\right]}^T,\hfill \end{array}$$

(20)

where $\mathbf{x}$ denotes the state vector, $\mathbf{u}$ is the input vector and $\mathbf{y}$ is the output vector. The EKF provides an estimation of the PMSM state from the difference between the measured output and calculated value. We define the transition matrix $\mathbf{F}$ using the Jacobian $\mathbf{F}={\left.\partial f/\phantom{\partial f\partial \mathbf{x}}\partial \mathbf{x}\right|}_{\widehat{\mathbf{x}}\left(t\right),\mathbf{u}\left(t\right)}$ as follows:

$$\mathbf{F}=\left[\begin{array}{cccc}-\frac{R_s}{L}& \omega & 0& 0\\ -\omega & -\frac{R_s}{L}& -\frac{\psi }{L}& 0\\ 0& \frac{3n_p^2\psi }{2J}& -\frac{b}{J}& 0\\ 0& 0& 1& 0\end{array}\right].$$

(21)

We set the observation matrix $\mathbf{H}$ using the Jacobian ${\mathbf{H}}_k={\left.\partial h/\phantom{\partial h\partial \mathbf{x}}\partial \mathbf{x}\right|}_{{\widehat{\mathbf{x}}}_{k|k-1}}$ as:

$$\mathbf{H}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right].$$

(22)

The EKF can be built by the following recursive relations:

$$\begin{array}{c}{\widehat{\mathbf{x}}}_{k|k-1}=\widehat{\mathbf{x}}\left(t_k\right),\hfill \\ {\mathbf{P}}_{k|k-1}=\mathbf{P}\left(t_k\right).\hfill \end{array}$$

(23)

where $\mathbf{P}$ is the error covariance of observations. In this phase, the state and error covariance are computed. The Kalman gain is calculated as follows:

$$\begin{array}{c}{\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T{\mathbf{S}}_k^{-1},\hfill \\ {\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T+{\mathbf{R}}_K.\hfill \end{array}$$

(24)

Then, the algorithm updates the estimation with measurement and error covariance as follows:

$$\begin{array}{c}{\tilde{\mathbf{y}}}_k={\mathbf{z}}_k-h\left({\widehat{\mathbf{x}}}_{k|k-1}\right),\hfill \\ {\mathbf{P}}_{k|k}=\left(\mathbf{I}-{\mathbf{K}}_k·{\mathbf{H}}_k\right)·{\mathbf{P}}_{k|k-1}.\hfill \end{array}$$

(25)

In the sensorless control of the PMSM, the EKF, as an estimator, provides the rotor speed and angle.

4.2. Gaussian Process Regression

The GPR is a non-parametric function regressor that uses Bayesian inference [35]. Using a training data set, the mean and covariance functions specify the model and estimate the system parameters. The GPR defined from the Gaussian distribution is expressed as follows:

$$\begin{array}{c}\mu \left(x\right)=E\left[g\left(x\right)\right],\hfill \\ k\left(x_i,x_j\right)=E\left[\left(g\left(x_i\right)-\mu \left(x_i\right)\right)\left(g\left(x_j\right)-\mu \left(x_j\right)\right)\right],\hfill \end{array}$$

(26)

where $\mu \left(x\right)$ is the mean function, and $k\left(·\right)$ is the covariance function. Thus, the GPR can be represented as follows:

$$g\left(x\right)\sim GP\left(\mu \left(x\right),k\left(x_i,x_j\right)\right).$$

(27)

Using a training set from observations, the GPR approximates the function $f\left(x\right)$. Based on the Bayesian regression method, the GPR assumes mean and covariance functions for modelling. In this study, the covariance function $k\left(x_i,x_j\right)$ was defined as a squared exponential kernel function:

$$k\left(x_i,x_j\right)=exp\left(-\frac{{∥x_i-x_j∥}^2}{2l^2}\right),$$

(28)

where l denotes the kernel function length. We obtained the training dataset from Q, R, and the EKF results. The dataset from the EKF exhibited a change in speed error and rotor current under variations in Q and R. GPR approximates the relationship between the covariance matrix and performance by training with the dataset. Thus, the GPR-EKF can choose the optimized parameters, Q and R, for improved performance in estimating the rotor speed.

5. Simulation

The PMSM speed control system was simulated using the MATLAB/Simulink software. The parameters of the PMSM are listed in

Table 1. Parameter specifications.

Symbol	Parameter	Value
$R_s$	Stator resistance	0.18 $\Omega$
$\psi$	Flux linkage	0.071 Wb
J	Inertia	0.6 $\times {10}^{-3}$ $\mathrm{kg}·{\mathrm{m}}^2$
p	Pole pairs	4
$K_e$	Back EMF constant	0.027 N·m/A
$K_t$	Torque constant	0.428 $V_{p}k$/krpm
$L_d$	Inductance of d-axes	0.835 mH
$L_q$	Inductance of q-axes	0.835 mH

.

The STRL-SMC and STRL-SMC with the WOA were compared in detail to verify the effectiveness of the proposed control methods. Figure 4 shows the results for STRL-SMC and STRL-SMC with the WOA. The reference speed was set to 300 rad/s. The rated speed and power for the PMSM used in simulation are 3000 rpm and 1.1 kW, respectively. The simulation result was made with a sampling time of 1 $\mathsf{\mu }$s, and the same sampling time was applied to both the current control loop and the speed control loop. The electrical and mechanical time constant in PMSM are ${\tau }_{elec}$ = 4.64 ms and ${\tau }_{mech}$ = 9.36 ms, respectively.

The parameters of WOA are given in

Table 2. WOA Parameters.

Parameter	Value
number of search agents	10
number of iterations	10
lower bound of variable	1000
upper bound of variable	80,000

. Figure 4 shows the case where the load torque was suddenly increased by 20 N·m at 0.04 s. For STRL-SMC with WOA, the system dynamics shows the reduced convergence time and the chattering in steady state becomes smaller than STRL-SMC about 60%, as shown in Figure 4. In

Table 3. Switching gains for STRL-SMC and STRL-SMC with WOA.

Parameter	Value
$k_1$	60,000
$k_2$	20,000
lower bound of initial $k_1$ value in WOA	50,000
upper bound of initial $k_1$ value in WOA	70,000
lower bound of initial $k_2$ value in WOA	10,000
upper bound of initial $k_2$ value in WOA	30,000

, the switching gains for STRL-SMC and STRL-SMC with WOA are presented. The switching gains of the STRL-SMC without WOA were chosen for the best performance. The gain $\alpha$ in Equation (7) was set to 1000. The values of $k_1$ and $k_2$ are randomly initialized in the range of parameters listed in Table 3. The variation range of $k_1$ and $k_2$ during optimization is indicated in Table 2. In STRL-SMC with the WOA, the system dynamics were faster and the chattering became smaller than those STRL-SMC, as shown in Figure 4. In Figure 4, the STRL-SMC without WOA takes 0.0382 s to reach the reference speed and 0.0148 s to recover speed after an external load is added at 0.04 s. For the STRL-SMC using WOA, the settling time is 0.0163 s. The STRL-SMC with WOA recovers the speed at 0.0420 s, which is 0.0128 s faster than STRL-SMC without the WOA. It is represented that the performance of STRL-SMC with WOA is more robust than STRL-SMC. Compared to STRL-SMC, the controller using WOA reduces speed tracking error in shorter time when the external load is added. The change of switching gains optimized by WOA are shown in

Table 4. WOA-SMC switching gains.

Time	${\mathit{k}}_1$	${\mathit{k}}_2$
0.005	80,000	79,984
0.01	80,000	79,984
0.015	79,983	55,473
0.02	9530	10,036
0.025	2758	4868
0.03	2381	4906
0.035	2744	4861
0.04	2389	4615
0.045	11,711	10,852
0.05	3045	5166
0.055	3351	5247
0.06	2677	4727

. The gains increased at 0.045 s with added external load. The speed response of the PMSM speed-tracking system is shown in Figure 4. Figure 5 shows the 3-phase current waveform under the same condition in Figure 4.

The reference speed presented in Figure 6 changed from 300 to 330 rad/s at 0.03 s. The reference speed decreased to 290 rad/s at 0.045 s. The switching gains that changed with WOA are listed in

Table 5. WOA-SMC switching gains.

Time	${\mathit{k}}_1$	${\mathit{k}}_2$
0.005	80,000	79,984
0.01	80,000	79,984
0.015	50,270	55,473
0.02	9530	10,036
0.025	2758	4868
0.03	2381	4906
0.035	11,223	9987
0.04	2377	4694
0.045	2959	5107
0.05	10,413	8741
0.055	2955	5003
0.06	2362	4631

. The variation of gains was related to the rotor speed change at 0.03 s and 0.045 s. It can be observed from Figure 6 that the speed fluctuations under a sudden reference speed change are different between STRL-SMC with WOA and STRL-SMC.

In Figure 6, the controller of STRL-SMC with WOA provides better tracking performance than the controller without WOA. The controller can track the reference speed with fast response and small error by using the WOA. In Figure 6, the settling time of STRL-SMC without WOA is 0.03 s while the proposed controller using WOA takes 0.0163 s. After the reference speed change at 0.03 s, the STRL-SMC without WOA tracks the speed at 0.0449 s. The settling time for the speed recovery of the controller without WOA is 0.0591 s after reference speed change at 0.045 s. The STRL-SMC using WOA takes 0.0005 s and 0.0014 s for speed recovery after the reference speed change at 0.03 s and 0.045 s, respectively. From Figure 7, it is represented that the three-phase current wave form is related to the reference speed change shown in Figure 6. The controller using the WOA proved the effectiveness in chattering reduction and dynamic performance. Figure 8 shows the PMSM speed tracking under an additional external load of 10 N·m at 0.04 s for a low speed of 30 rad/s. In Figure 9, the reference speed increases from 30 rad/s to 40 rad/s at 0.03 s. Moreover, at 0.045 s, the reference speed changes to 25 rad/s. Figure 10 presents the error in the estimated speed from the EKF and the measured speed. Figure 10 shows that the estimated speed from the GPR-EKF exhibits fewer oscillations than the EKF only. The error in the rotor angle between the EKF estimation and measurement is shown in Figure 11. It is observed that the GPR-EKF has less error than the EKF only. With the covariance matrix tuning of the GPR, the EKF algorithm can track the rotor speed with accuracy and robustness.

6. Conclusions

This paper describes the sensorless control of a PMSM using an SMC and an EKF. The proposed method is designed for chattering reduction and fast speed tracking. The WOA was used to design the switching gain for the SMC. The gain in SMC directly affects the motor control effectiveness including chattering and disturbance rejection. Using WOA, the chattering phenomenon caused by the large switching gain was reduced, and speed-tracking performance was improved.

In sensorless control, the accuracy of PMSM state estimation and elimination of error between speed value from PMSM and estimator are the key issue. The GPR-EKF updated the rotor state with input data of current and voltage data and provided the data to the controller. GPR was applied to the EKF to estimate the rotor speed and angle accurately. The GPR-EKF functioned as a rotor speed estimator, and the WOA-SMC was successfully used for speed control. Comparative simulations indicate that the proposed SMC has a better performance than that without optimization. The effectiveness of the covariance matrix tuned by the GPR was demonstrated through simulations. The optimized controller can improve the performance of a sensorless PMSM control in real industrial applications. Future research can focus on experimental study of the proposed controller operation. For this purpose, a real-time implementation of the method will be further investigated. Furthermore, to guarantee the proposed controller performance, reliable PMSM modelling, and measurement data are required. The appropriate design of the WOA cost function is also needed. The WOA-SMC and GPR-EKF can be further developed and applied to industry development.