Machine learning techniques for vector control of permanent magnet synchronous motor drives

Abstract

In the conventional vector control technique for motor drive, Proportional-Integral (PI) controllers are being used, which are sensitive to parameter variations of the drive system. This article presents Machine Learning (ML)-based controllers for a surface permanent magnet synchronous motor (PMSM) drive system. In this work, ML-based regression algorithms such as linear regression, support vector machine regression and feedforward neural network are investigated for speed control application. The entire vector control scheme implementing the ML-based control algorithms is investigated theoretically and simulated under various dynamic operating conditions. Simulation results and performance metrics are compared with those of the conventional PI controller, and they validate the effectiveness of the proposed control algorithms for speed control applications. The proposed ML-based controllers have the ability to meet the speed tracking requirements in the closed-loop system, with performance metrics superior to those of the PI controller, by an average value of 20% for different test scenarios. The transient levels of the motor drive reduce by 0.02% while using the proposed controllers.

Keywords:

• Controllers
• machine learning (ML)
• neural network
• permanent magnet synchronous motor (PMSM)
• regression algorithms
• vector control

REVIEWING EDITOR:

• Ai Qingsong, Wuhan University of Technolog, Wuhan, China

SUBJECTS:

• Technology
• Computer Science
• Algorithms & Complexity
• Artificial Intelligence
• Engineering & Technology
• Electrical & Electronic Engineering
• Systems & Control Engineering
• Transport & Vehicle Engineering

1. Introduction

Permanent magnet synchronous motors (PMSM) are a suitable option for the automobile sector because of their efficiency, compactness and dependability (Raia et al., 2021; Sorial et al., 2021). Durability, high power density, prolonged lifetime, less maintenance and low noise are some of the other benefits. However, the cost of the motor is higher compared to its peers (Jung et al., 2015; Pillay & Krishnan, 1989; Qutubuddin & Yadaiah, 2018; Rind et al., 2017; Soliman, 2019). The motor current control methods, as well as the hardware design, determine the overall performance of a PMSM drive (Li et al., 2020; Liu & Luo, 2017). As far as control technique is concerned, direct torque control, V/f control, vector control and artificial intelligence (AI)-based control are some of the key control methodologies for the motor.

The V/f control technique is affordable, easy to implement and can be used in both open-loop and closed-loop applications. However, it gives poor performance at low speeds. The vector control is the most commonly used control technique and can be used over larger speed variations, starting from low speeds (Hannan et al., 2018; Rind et al., 2017). It employs either Proportional-Integral (PI) or Proportional-Integral-Derivative (PID) controllers to perform the control operation. The designing procedure and practical implementation are simple for these controllers. However, their gain values are fixed and they are susceptible to system uncertainties and require an accurate mathematical model to determine controller gains (Jung et al., 2015; Qutubuddin & Yadaiah, 2017).

On the other hand, AI-based controllers can be used even if the system is ill-defined and can be trained using sufficient data. The Adaptive Neural Fuzzy Inference System, Artificial Neural Network and Genetic Algorithm are some of the AI control methods used for control applications (Hannan et al., 2018). With the emergence of graphics processing units, space constraints and the training-learning method of AI-based controllers, which are time-consuming, have been addressed (Karim et al., 2019).

Machine learning (ML) is a category of AI that can convert knowledge into expertise or uncover useful patterns in enormous volumes of data. This technique uses domain knowledge rather than mathematical modeling to train and produce an ML model (Jaffar et al., 2020; Morais & Pedro, 2018). ML approaches are broadly classified as unsupervised, semi-supervised, supervised and reinforcement learning. Supervised learning is further divided into regression and classification techniques. Regression techniques are for continuous output problems while classification is for discrete output problems (Dahrouj et al., 2021; Daliya et al., 2021; Simeone, 2018). Some of the most popular ML regression methods are decision trees, linear regression (LR), neural networks (NN), random forests and support vector machines (SVMs) (Farsi et al., 2021; Mahmud et al., 2021).

The main benefit of the regression techniques is that they perform better on smaller datasets (Jaffar et al., 2020), but NNs can also learn intricate correlations between features and targets (Hannan et al., 2018). The speed control of a motor drive can be modeled as a regression problem. Some of the aforementioned regression approaches are examined in this work for speed control of PMSM drive.

The following are the key goals of this article:

1. To design a current controller based on ML that uses less complex circuitry.

2. To investigate the possibility of multiple ML regression approaches for effectively controlling the PMSM motor drive used in electric vehicles (EV).

3. To propose an alternative to the conventional controller utilized in industry, from the developing field of AI.

This article is organized as follows: Section 2 discusses the theoretical aspects of PMSM drive, whereas Section 3 describes the conventional vector control approach. Section 4 describes the proposed method as well as the ML algorithms used. Section 5 analyzes the validity of the proposed AI model using several conditions, and Section 6 presents the conclusion.

2. PMSM model

For a PMSM drive, its mathematical model has an electrical component and a mechanical component (Butt & Rahman, 2013; Li et al., 2020; Qutubuddin & Yadaiah, 2017; Sorial et al., 2021). Equation (1)(1) $$\left(\begin{array}{c}{v}_{\mathit{\text{sd}}}\\ v_{\mathit{\text{sq}}}\end{array}\right)=\left(\begin{array}{cc}{R}_s+L_d\frac{d}{\mathit{\text{dt}}}& -w_e{L}_q\\ w_e{L}_d& R_s+L_q\frac{d}{\mathit{\text{dt}}}\end{array}\right)\left(\begin{array}{c}{i}_d\\ i_q\end{array}\right)+\left(\begin{array}{c}0\\ w_e{\psi }_f\end{array}\right)$$(1) provides the electrical component of the synchronously rotating coordinate system (dq-axis). (1) $$\left(\begin{array}{c}{v}_{\mathit{\text{sd}}}\\ v_{\mathit{\text{sq}}}\end{array}\right)=\left(\begin{array}{cc}{R}_s+L_d\frac{d}{\mathit{\text{dt}}}& -w_e{L}_q\\ w_e{L}_d& R_s+L_q\frac{d}{\mathit{\text{dt}}}\end{array}\right)\left(\begin{array}{c}{i}_d\\ i_q\end{array}\right)+\left(\begin{array}{c}0\\ w_e{\psi }_f\end{array}\right)$$(1) where the stator resistance is given by R_s, ω_e is electrical motor speed; dq-axes instant stator voltage are given by v_sd, v_sq; i_d, i_q are d and q components of instant stator current; L_d, L_q are the stator and rotor d- and q-axis inductances; and ψ_f is the flux linkage produced by the permanent magnet (PM).

The mechanical component of the PMSM drive is given by the following torque equations: (2) $$\mathit{\text{Te}}=J_{\mathit{\text{eq}}}\frac{d{w}_m}{\mathit{\text{dt}}}+B_a{w}_m+T_L$$(2) where J_eq is the inertia of the motor, ω_m is the motor rotational speed, B_a is the friction coefficient, T_L is load torque and T_e is electromagnetic drive torque. (3) $$T_e=\frac{3P}{2}\left[{\psi }_f{i}_q+\left(L_d-L_q\right)i_d{i}_q\right]$$(3) where P is the pole pairs in the motor. L_d = L_q in a surface PMSM type motor.

Equations (4)(4) $$w_e=w_m\mathrm{*}P$$(4) and (5)(5) $${\theta }_e=\int w_e\mathrm{ }\mathit{\text{dt}}$$(5) give the relations between motor speeds and rotor angle θ_e. (4) $$w_e=w_m\mathrm{*}P$$(4) (5) $${\theta }_e=\int w_e\mathrm{ }\mathit{\text{dt}}$$(5)

3. Conventional vector control

For PMSM drives, vector control method is referred to as field-oriented control and recommended due to its superior performance across a broad speed range. The stator current is divided into torque and flux current components using this method. Figure 1 schematically depicts this control method used in the PMSM motor drive.

Figure 1. Conventional vector control.

By rearranging Equations (1)–(3), the transfer functions needed to design the PI controllers are obtained. The following is the transfer function that can be used to obtain the controller parameters in a surface PMSM drive: (6) $${\mathit{\text{TF}}}_w=\frac{{\psi }_f.P}{\left({\mathrm{ }B}_a+s.J_{\mathit{\text{eq}}}\right)}$$(6)

During the control process, the compensation terms (–w_eL_qi_q, w_eL_di_d, w_eΨ_f) are employed for fully decoupling the stator current into the torque and flux current components. But, they are ignored while designing the current-loop PI controllers. Equations (7)(7) $${\mathit{\text{TF}}}_{\mathit{\text{id}}}=\frac{1}{\left({\mathrm{ }R}_s+s.L_d\right)}$$(7) and (8)(8) $${\mathit{\text{TF}}}_{\mathit{\text{iq}}}=\frac{1}{\left({\mathrm{ }R}_s+s.L_q\right)}$$(8) give the transfer functions for these controllers. In the final stage of the current loop, the compensation term needs to add with the outputs of the current. This prevents errors brought on by decoupling (Brejl & Princ, 2012; Hu et al., 2019; Li & Liu, 2019; Li et al., 2020; Liu & Luo, 2017). The gains for the PI controllers are obtained by Ziegler–Nicholas tuning method in the conventional technique (Qutubuddin & Yadaiah, 2017). (7) $${\mathit{\text{TF}}}_{\mathit{\text{id}}}=\frac{1}{\left({\mathrm{ }R}_s+s.L_d\right)}$$(7) (8) $${\mathit{\text{TF}}}_{\mathit{\text{iq}}}=\frac{1}{\left({\mathrm{ }R}_s+s.L_q\right)}$$(8)

4. Proposed ML-based control

4.1. ML-based control technique

Figure 2 displays a vector control method based on ML regression techniques for the PMSM drive. In the proposed method, the current loop uses ML-based controllers instead of PI controllers. However, the speed loop remains the same (Li et al., 2020; Qutubuddin & Yadaiah, 2017).

Figure 2. Proposed ML-based control method.

The steps to design the proposed controller is as follows: (i) modeling the drive system (ii) ML model comparable to the conventional controller dynamics and (iii) training of the ML controller (Li et al., 2020). Figure 3 outlines the structure for training and validating the ML models.

Figure 3. Proposed ML model framework.

4.2. Training the ML controllers

Deep Learning Toolbox^TM in MATLAB is used to train the ML models. A 600,000-sample training dataset is derived using a conventional model with PI current controllers, which is operated under the rated conditions, as

well as in the presence of load disturbances and speed changes. Together with a few independent samples, a portion of the dataset is used to test the model. The test sample size ranges between 50,000 and 200,000.

The ML block receives the dq-axes current error terms and their integrals as input signals. There is no need to process the compensation terms further because they are taken into consideration when determining outputs from ML model (Li et al., 2020). Consequently, this lowers the control algorithm’s complexity. When used in a PMSM drive, the trained ML controller can easily replace and operate similarly to a PI controller.

For estimating dq-axes reference voltages, LR and SVM algorithms each require two models, while a single model can be used for NN. This is because LR and SVM are single output networks, while the NN model can be a multi-output model. Figure 4 depicts the proposed ML model. Table 1 gives the tuned parameters for each ML model. The hyperparameters for each ML model are determined by manually experimenting with various parameter combinations (from the options in the Toolbox) and then validating with the test dataset. The number of neurons and hidden layers in the NN model are determined by a trial-and-error process.

Figure 4. Layout of proposed ML model.

Table 1. Hyperparameter settings for ML models.

ML model	Hyper-parameter	Selection
LR	Learner	svm
	Solver	BFGS
	Regularization	Ridge
	Lambda	1 / (Training sample size)
SVM	Kernel function	Linear
	Kernel scale	1
	Solver	SMO
NN	No. of hidden layers	2
	Size of hidden layers	10 + 2
	Activation function	Sigmoid function (hidden layer), linear function (output layer)
	Training algorithm	Levenberg-Marquardt algorithm

4.2.1. Linear regression

The linear regression (LR) method is one of the most commonly used algorithm that represents the correlation between data by drawing a line through the data with the least value of Mean Square Error loss (Alquthami et al., 2022; Farsi et al., 2021; Gutierrez-Gomez et al., 2020; Mahmud et al., 2021). In Equation (9)(9) $$y={\theta }_0+{\sum }_{i=1}^N{\theta }_i{x}_i$$(9) , y represents the output of the LR model, x_i represents the input; θ_i is the weight of each ith input, θ₀ is the bias; and N represents the number of features in the dataset (Jaffar et al., 2020). (9) $$y={\theta }_0+{\sum }_{i=1}^N{\theta }_i{x}_i$$(9)

The function ‘fitrlinear’ is used for training LR regression models in MATLAB. The parameters used are as follows:

• (i)‘Learner’:

  The ‘Learner’ is a LR model type that can be set as LR via ordinary least squares (‘leastsquares’), or Support vector machine regression (‘svm’).

• (ii)‘Solver’:

  The ‘Solver’ or objective function minimization technique can be Stochastic gradient descent (SGD), Average stochastic gradient descent (ASGD), Dual SGD, Broyden–Fletcher–Goldfarb–Shanno quasi-Newton algorithm (BFGS), Limited-.

memory BFGS (LBFGS) or Sparse Reconstruction by Separable Approximation (SpaRSA) method.

• (iii)‘Regularization’:

  ‘Regularization’ or complexity penalty type can be set as Lasso or Ridge.

• (iv)‘Lambda’:

  ‘Lambda’ is also known as the Regularization term strength and is calculated as the inverse of the training sample size.

After experimenting with different combinations for Learner-Solver-Regularization, the optimum combination is found to be BFGS-svm-Ridge.

4.2.2. SVM regression

A kernel technique is used in this method to convert the independent variables into a higher-dimensional feature space. This method requires less computing power, is independent of the dimensionality of the input, and seeks to minimize generalization mistakes rather than training faults (Gutierrez-Gomez et al., 2020; Jaffar et al., 2020; Pirbazari et al., 2021).

Equation (10)(10) $$y={\sum }_{i=1}^N\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right).〈x_i,x〉+b$$(10) is applied to a dataset that is linearly separable, and Equation (11)(11) $$y={\sum }_{i=1}^N\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right).K\left(x_i,x\right)+b$$(11) is applied to a dataset that is nonlinearly separable (Mahmud et al., 2021). x_i represents the inputs and y represents the outputs. α_i represents the contraction coefficient and b represents the bias. K(x,y) represents the kernel function. (10) $$y={\sum }_{i=1}^N\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right).〈x_i,x〉+b$$(10) (11) $$y={\sum }_{i=1}^N\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right).K\left(x_i,x\right)+b$$(11)

The ‘fitrsvm’ function in MATLAB is used to train SVM regression models. The following parameters must be given when training the SVM model:

• (i) Kernel function:

  Gaussian, Linear, Radial Basis Function (RBF) and Polynomial kernels are the available options.

• (ii) Kernel scale:

  This value can be 1. Otherwise, it is set as auto where a subsampling procedure in the software decides the value.

• (iii) ‘Solver’:

  Iterative Single Data Algorithm (ISDA), L1 Quadratic Programming (L1QP) or Sequential Minimal Optimization (SMO) can be used as the optimization procedure or Solver.

The optimal result arises from configuring SMO for ‘Solver’, scale value as 1, and opting for linear kernel function. The linear kernel function is represented by (12) $$K\left(x,y\right)=x^{\prime }y$$(12)

4.2.3. Feedforward NN

A feedforward NN is a type of artificial NN that only allows information to flow from the input to the output. The artificial neurons of NNs are modeled after biological brains, and structured into three levels: input, output and hidden layers between the two. The depth of the network is determined by the number of hidden layers. Each layer has a bias, while each connection between neurons is represented by a weight (Alquthami et al., 2022; Farsi et al., 2021; Mahmud et al., 2021; Shalev-Shwartz & Ben-David, 2014; Zhang et al., 2019).

Figure 5 shows the architecture of a two-input single-output NN having one hidden layer with three neurons. Here, x₁, x₂ are the inputs, and y is the output. As seen in Figure 6, the output of an artificial neuron is a function f of the sum of bias b and all the weights (w₁–w_n) from inputs (x₁–x_n) (Jaffar et al., 2020; Karim et al., 2019; Morais & Pedro, 2018). Activation function f, given by Equation (13)(13) $$y=f\left(b+{\sum }_{i=1}^n{x}_i{w}_i\right)$$(13) , is used for normalizing the output of a neuron. (13) $$y=f\left(b+{\sum }_{i=1}^n{x}_i{w}_i\right)$$(13)

Figure 5. NN architecture.

Figure 6. Layout of an artificial neuron.

To train a simple NN in MATLAB, the ‘train’ function is used with the following parameters specified.

• (i) Number of hidden layers and Size of each hidden layer:

  A trial-and-error method is followed to select the number and size of the hidden layers, starting from the minimum values of 1 and 4 (input variable size), respectively. A satisfactory result is obtained at 2 layers, with 10 and 2 neurons.

• (ii) Activation function of each layer:

  A hyperbolic tangent sigmoid transfer function is set as the hidden layer activation function while the output layer has a linear activation function.

• (iii) Training algorithm:

  Levenberg–Marquardt, Bayesian Regulation, Gradient descent and Resilient back propagation (RPROP) algorithms are some of the training algorithms for NNs in MATLAB, among which the Levenberg–Marquardt algorithm is the fastest and most popular.

4.2.4. Training procedure

Each ML algorithm is prepared for training as mentioned in subsections 4.2.1–4.2.3. After that, they undergo offline training for regression from sequence to sequence. The trained models are verified using a small number of samples from the training data set. A comparison of the PI outputs and the

reference voltage outputs of the ML models is presented in Figure 7. The transient peaks of the controller outputs are lesser for the proposed ML controllers. LR-based and SVM-based controllers have the least maximum transient peak values and the voltage waveforms settle similar to that of the PI controller.

Figure 7. Conventional vs. ML-based controller performance: dq-axis reference voltages: (a) v_sd-ref and (b) v_sq-ref.

5. Performance evaluation of controllers

MATLAB is used in the development of the PMSM motor drive simulation model. Table 2 lists the parameters for the PMSM drive (Qutubuddin & Yadaiah, 2017, 2018). The controller sampling rate is 1 µs. Using Simulink, the training data set from the traditional model is first created. Performance analysis is then conducted using the models created by MATLAB Coder for both the proposed and traditional approaches. The testing of the trained ML models is detailed in subsections 5.1–5.5 (Li et al., 2020; Qutubuddin & Yadaiah, 2017, 2018).

Table 2. PMSM parameters in Simulation model (Qutubuddin and N. Yadaiah, 2018, 2017).

Parameter	Description	Value	Units
R	Resistance	2.85	Ω
Ld, Lq	Inductance	0.0085	H
Ba	Friction coefficient	1 e-4	N.m-s
Jeq	Inertia	8 e-4	kg-m^2
P	Pole pairs	4	–
Ψf	Mutual flux linkage	0.1548	Wb-turns
ωm	Rated speed	300	rad/s

5.1. Current control evaluation

While obtaining the d- and q-axes currents, the test motor’s speed is maintained constant to evaluate the current-loop controllers. The applied load torque is set to 5 N.m., and the required motor speed is selected as 300 rad/s (Qutubuddin & Yadaiah, 2017, 2018). The responses of the conventional and ML-based controllers are shown in Figures 8–11.

Figure 8. Conventional vs ML-based controllers: Motor speed response.

Figure 9. Conventional vs ML-based controllers: Motor torque response.

Figure 10. Conventional vs ML-based controllers: Current response of motor drive.

Figure 11. Conventional vs ML-based controllers: dq-axis currents (i_d, i_q) of PMSM drive.

The PI controller’s speed response (Figure 8) settles to 300 rad/s at 0.017 s with an overshoot of 1.15%. In contrast, the speed responses of the LR-based and SVM-based controllers exhibit a peak overshoot of 1.13% and settle to the required speed at 0.02 s. A 1.17% peak overshoot is seen by the NN-based controller when it settles to 300 rad/s at 0.022 s (Table 3).

Table 3. Comparison of speed response under rated conditions (Figure 8).

Description	PI	LR	SVM	NN
Peak overshoot (%)	1.15	1.13	1.13	1.17
Settling time (s)	0.017	0.02	0.02	0.022

Figure 9 shows how each controller’s torque responses reach the rated load (5 N.m.), with the PI controller responding the fastest. NN-based controllers have the lowest maximum torque value (93 N.m.) during the transient state compared to PI controllers (97 N.m.), LR-based and SVM-based controllers (101 N.m.).

The three-phase motor current responses of ML-based and traditional controllers are displayed in Figure 10. The PI controller produces a rated current of 6 A at 0.02 s and the ML-based controllers at 0.035 s. The transient state peak values of the LR-based and SVM-based controllers (80 A) are lesser than that of the PI and NN-based controllers (100 A). The dq-

axis currents of the conventional and proposed models are compared in Figure 11. At 0.01 s, the d-axis current of a PI controller settles to 0 A, while the q-axis current settles to 5.5 A, at 0.02 s. The LR-based and SVM-based controllers reach the same current values at 0.025 s and 0.022 s, respectively, while the NN-based controller takes 0.06 s and 0.03 s, respectively (Table 4). The NN model has more current ripples in the dq-axis current responses, compared to that of the other controllers.

Table 4. Comparison of current control performance (Figure 11).

Settling time	PI	LR	SVM	NN
For id = 0 A	0.01	0.025	0.025	0.06
For iq = 5.5 A	0.02	0.022	0.022	0.03

5.2. Speed control evaluation

PMSM speed tracking is used to assess the effectiveness of the controllers. The test starts with 10% of desired speed (30 rad/s), increased to 50% (150 rad/s), then 100% (300 rad/s) and finally 125% (375 rad/s) at time intervals of 0.05 s each.

As seen in Figure 12, the speed responses with the PI controller and the NN-based controller are smoother than the speed responses with the LR-based and SVM-based controllers at very low speeds (10%). All controllers respond similarly at half the rated speed. The responses of the LR-based and SVM-based controllers are similar to that of a conventional controller for speeds over rated speed (100–125%), but slightly sluggish. The NN-based controller gives a smooth response at higher speeds.

Figure 12. Conventional vs ML-based controllers: Motor speed response.

Figures 13 and 14 illustrate how the changes in motor 3-phase currents and torque correspond with variations in speed. For speeds over half the rated speed, torque transients are lesser for the LR and SVM models, and current responses are the same as that of the conventional model. At speeds below the rated speed, the NN model gives the same responses as the conventional model. For rated speed and above, the NN model has more oscillations in the torque response.

Figure 13. ML-based vs. Conventional controllers: Torque response for speed variations.

Figure 14. ML-based vs Conventional controllers: Motor current response for speed variations.

5.3. Effect of load disturbance

A 2.5 N.m. continuous load is applied at a time step of 0.03 s after the motor is started at no load, and then another 2.5 N.m. is applied at a time step of 0.07 s. The motor drive performances with both conventional and ML-based controllers are shown in Figures 15–17, with the load being the only parameter that differs from Table 2. The torque responses of the PI controller and the proposed controllers in relation to the additional load are shown in Figure 15. Figure 16 illustrates how the speed decreases with the introduction of a load and how fast the controllers recover it. Figure 17 illustrates the response of the 3-phase current variations correspond to variations in load.

Figure 15. ML-based vs. Conventional controllers: Torque response for load disturbances.

Figure 16. ML-based vs. Conventional controllers: Speed response for load disturbances.

Figure 17. ML-based vs. Conventional controllers: Motor current response for load disturbances.

5.4. Robustness of speed loop controller

Based on the road conditions and load on the EV, motor inertia and friction factor can change, in practical cases. This will have an impact on the output of the speed loop controller.

Figure 18 compares the conventional and ML-based vector-control methods when the friction factor is doubled and the inertia is tripled, while the other conditions are the same as in Table 2.

Figure 18. ML-based vs. Conventional controllers: Motor speed response for larger J_eq and B_a.

In comparison to Figure 8, speed responses of PI, LR-based and SVM-based controllers reach the rated speed at 0.08 s while the NN-based controller takes 0.1 s. This is because of higher inertia and friction factors and holds for all the controllers.

5.5. Evaluation metrics

To fully capture all the features of prediction performance that matter to applications, a single metric is insufficient. Hence, different metrics are considered and the model with the majority of better metrics is determined as the best one. The performance of the trained ML algorithms is assessed using the following metrics: symmetric mean absolute percentage error (SMAPE), mean absolute error (MAE) and root mean square error (RMSE). These are statistical metrics that are frequently used to evaluate how well time series-based models work. For these metrics, a smaller error indicates better performance of the model.

• (i) MAE

  MAE calculates the absolute value of the difference between the observed and predicted values, without taking into account the direction of the errors. As demonstrated in Equation (14)(14) $$\mathrm{\text{MAE}}=\frac{1}{N}{\sum }_{i=1}^N\left|\left({\overline{y}}_i-y_i\right)\right|$$(14) , it provides a quantitative measure of the errors by averaging the errors throughout the entire group and is utilized as a loss function for regression-based ML algorithms. (14) $$\mathrm{\text{MAE}}=\frac{1}{N}{\sum }_{i=1}^N\left|\left({\overline{y}}_i-y_i\right)\right|$$(14)

• (ii) RMSE

  In supervised learning applications, RMSE is a commonly used metric to assess the accuracy of predictions. It demonstrates how well accepted the observations-based predictions are. A lower RMSE value represents a lesser predicting error. RMSE refers to the standard deviation of the variations between the observed (actual) values and the anticipated values. (15) $$\mathrm{\text{RMSE}}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left({\overline{y}}_i-y_i\right)}^2}$$(15)

• (iii) SMAPE

  SMAPE is a tool used for technique evaluation and comparison. This relative error metric allows for cross-series comparisons as well as comparisons with the outcomes of the forecasting methods that produced the data sets. To provide an idea of the variability across the various time series, the standard deviation for each approach is provided. (16) $$\mathrm{\text{SMAPE}}=\frac{100\mathrm{\%}}{N}{\sum }_{i=1}^N\frac{\left|\left(\overline{y_i}-y_i\right)\right|}{\left(\left|\overline{y_i}\right|+\left|y_i\right|\right)/2}$$(16)

where y_i and ȳ_i are the actual and estimated values, respectively; N is the test sample size (Ahmadi et al., 2020; Asthana et al., 2023; Bao et al., 2014; Duan & Kashima, 2021; Jawad et al., 2020; Lemke & Gabrys, 2010; Martínez & Rocha, 2023; Méndez et al., 2023; Naz et al., 2023; Varoquaux & Colliot, 2023).

The assessment metrics of the drive speed are displayed in Table 5 and Figure 19 for the following scenarios: for rated load conditions (Figure 8), when the motor speed is increased from low speed to above rated speeds (Figure 12), and when load perturbations are applied (Figure 16). In all the three test cases, the PI controller gives the least values of all the errors. Subsections 5.5.1–5.5.3 give the evaluation metrics for reference voltage outputs, i.e. the direct outputs of the controllers, under different conditions. The least error in each set is highlighted in bold.

Figure 19. Metrics comparison of speed response (Table 5).

Table 5. Evaluation metrics for speed response under rated conditions.

	Metrics	PI	LR	SVM	NN
Rated conditions	MAE	10.9072	11.2512	11.1903	11.9194
	RMSE	40.6125	40.8831	40.8455	40.9932
	SMAPE	5.0976	5.2276	5.2072	5.4389
Speed variations	MAE	7.5233	10.0333	9.1848	8.3171
	RMSE	18.2025	19.7951	19.3128	19.0128
	SMAPE	7.9562	15.6132	12.1241	8.2194
Load disturbances	MAE	12.5194	12.6716	12.6216	15.3829
	RMSE	40.3287	40.8510	40.8211	41.0977
	SMAPE	5.5002	5.5565	5.5402	6.4487

5.5.1. Rated conditions

Table 6 and Figure 20 show the evaluation metrics for reference voltage outputs from all the controllers under rated conditions (Figure 7). The SVM-based controller has the least values of MAE and RMSE while the PI controller has the least SMAPE, when the reference voltage outputs are compared.

Figure 20. Metrics comparison of reference voltage output under rated conditions (Table 6).

Table 6. Evaluation metrics (v_sd-ref, v_sq-ref)—rated conditions.

	Metrics	PI	LR	SVM	NN
vsd-ref	MAE	23.444	21.888	21.519	25.564
	RMSE	79.294	55.034	54.870	64.101
	SMAPE	20.733	27.639	27.521	30.166
vsq-ref	MAE	32.084	31.447	31.159	42.235
	RMSE	153.166	138.953	137.574	163.905
	SMAPE	7.216	7.466	7.415	12.048

5.5.2. Speed variations

Table 7 and Figure 21 give the evaluation metrics of dq-axis reference voltages during the speed variation test. The proposed SVM-based controller exhibits least RMSE when compared with the PI controller

Figure 21. Metrics comparison of reference voltage output under speed variations (Table 7).

Table 7. Evaluation metrics (v_sd-ref, v_sq-ref)—speed variations.

	Metrics	PI	LR	SVM	NN
vsd-ref	MAE	15.0201	21.7482	19.9905	18.3291
	RMSE	44.5536	41.6567	40.3158	46.2872
	SMAPE	25.2853	58.2353	54.4894	31.9344
vsq-ref	MAE	14.9334	22.6870	19.9989	20.2859
	RMSE	53.3610	53.7285	51.3261	60.3123
	SMAPE	8.3370	30.3406	23.1005	10.8283

has the least values of MAE and SMAPE, when the reference voltage outputs are compared.

5.5.3. Load disturbances

Table 8 and Figure 22 give the evaluation metrics of dq-axis reference voltages during the load disturbance test. The SVM-based controller exhibits the least RMSE while the conventional PI controller has the least values of MAE and SMAPE, when the reference voltage outputs are compared.

Figure 22. Metrics comparison of reference voltage output under load variations (Table 8).

Table 8. Evaluation metrics (v_sd-ref, v_sq-ref)—load disturbances.

	Metrics	PI	LR	SVM	NN
vsd-ref	MAE	25.641	28.041	27.498	40.942
	RMSE	83.990	66.513	66.205	77.545
	SMAPE	18.567	26.414	26.164	44.119
vsq-ref	MAE	32.951	33.323	33.099	60.823
	RMSE	155.075	141.378	140.103	171.853
	SMAPE	8.422	9.426	9.399	27.867

5.6. Results and discussion

The proposed ML-based controllers (LR, SVM and NN) are compared with the PI controller under various test scenarios. The reference voltage outputs of the controllers are used in conjunction with the motor current, torque and speed responses in the evaluation. From the graphs, it is clear that for speeds below rated speed, an NN-based controller is a good alternative for the PI controller. For speeds at and above the rated speed, an LR-based or an SVM-based controller makes the better choice.

Under the rated conditions, the responses of the LR-based and SVM-based controllers are almost similar to those of the PI controller. All the ML-based controllers respond similar to the PI controller, during the load disturbances. The transient state peaks are reduced by 0.02% when ML-based controllers are used.

When the electric supply is switched on, it results in a switching transient. Hence, the magnetic flux in the motor and the corresponding motor current undergo a transient state before reaching the steady-state values. With increasing time, this transient response decays to 0, and only the steady-state response remains. The transient state is a short-term response, while the steady state is a long-term response that is more important (Hambley, 2019) to assess the performance of a motor. Figures 7–11 and 18 in this article show this transient state at the start of the torque, speed and current responses. Transient state occurs even when the reference speed or load applied to the motor changes (Figures 12–17). The transient state values vary for different values of speed or load, which cannot be predicted accurately by the conventional controller or the ML-based controllers. This is given by Figures 12–18.

The direct outputs of the controllers are the reference voltages v_sd and v_sq. The transient state values of the ML-based controller outputs had less undershoots and overshoots during rated conditions and load variations than the PI controller outputs, resulting in improved metrics. The oscillations in the LR and SVM-based controller responses during low-speed transients cause higher errors in the speed variation testing. However, motor drives in EV applications are not frequently required to run at such low speeds. The range of the dataset can affect the MAE and RMSE values, as they are range-dependent metrics. As a percentage metric, SMAPE can be affected by large variations in the observed and predicted values during the transient state. Regardless of the metric type used for performance evaluation, all their values are evaluated, taking into account all of these factors. The best model is determined to be the controller with the maximum number of improved metrics. The SVM-based controller is chosen as the best alternative for the PI controller based just on the metrics of the controller outputs.

Under all test conditions, the SVM-based controller gives the least RMSE for the reference voltages, in comparison to the PI controller and LR-based and NN-based controllers. The SVM-based controller also gives the least MAE for reference voltages under rated conditions. The PI controller has the least SMAPE in all the cases, and the least MAE during speed and load variations.

However, the evaluation of the overall drive response gives a different outcome. In the case of the drive speed response, the PI controller has the least values in all the errors, under all test conditions. The leading margin for PI is by 9% (NN), 10% (SVM) and 16% (LR), due to low-speed oscillations, and due to the inclusion of the compensation terms on the output side of the PI controller. The compensation terms (–w_eL_qi_q, w_eL_di_d, w_eΨ_f) are time-dependent functions of the dq-axis motor currents and electrical speed; which are added to the PI controller outputs to avoid decoupling inaccuracy while implementing the vector control technique. The instantaneous values of the compensation terms help the PI controller give better drive responses, even though ML-based controllers give better metrics.

The training data set for the ML models are inclusive of the compensation terms indirectly. The ML models are designed such that the controller outputs given to the next stage in the drive are independent of the compensation terms. These ML-based controllers are able to attain results nearer to that of a conventional controller that needs the compensation terms. This could indicate that with further refinement the ML models, in particular the SVM-based controller, can give improved performance for the overall drive response.

6. Conclusion

This article presented a vector-controlled PMSM drive based on ML algorithms. The main objective is to explore the feasibility of applying ML in the speed control of motor drives. Under various test situations, the proposed controller’s performance is compared to that of the conventional controller. The ML-based controllers could reproduce the conventional controller responses to some extent, without depending on the compensation terms.

The performance metrics of the ML-based controllers, especially RMSE, are superior when compared with conventional PI controller. Further, the RMSE of the SVM-based controller are superior to those of the PI controller by 20% and 8% for the respective controller output voltages during all the tests, followed by the LR-based controller in the same range. This indicates both controllers as good alternatives to the PI controller. At low speeds, the NN-based controller mimics the PI controller performance, making it a good alternative for low-speed requirements.

However, the overall drive response of the PI controller has better metrics due to the constant presence of compensation terms, by an average value of 12%. The performance of the proposed controllers can be further improved by fine-tuning the training parameters and with a bigger data set. Future work on this topic will include improvements to the ML designs to attain lower errors by factoring in the impact of instantaneous values of the compensation terms, and better performance of the motor drive.

Disclosure statement

The authors declare that they have no conflict of interest.

Data availability statement

The data that support the findings of this study are available from the corresponding author, J. L. Febin Daya, upon reasonable request.

Additional information

Notes on contributors

Ashly Mary Tom

Ashly Mary Tom is a doctoral student at the School of Electrical Engineering, Vellore Institute of Technology, Chennai Campus, India. Her research focuses on motor drives and machine learning techniques.

J. L. Febin Daya

J.L. Febin Daya is presently serving as Professor in Electric Vehicles Incubation, Testing, and Research Center at Vellore Institute of Technology, Chennai Campus, India. His current research interests include electrical drives and control, wireless charging, intelligent systems, electric vehicles and precision agriculture.