Multi-Objective Optimization of PMSM Servo System Control Performance Based on Artificial Neural Network and Genetic Algorithm

Abstract

With the rapid advancement of intelligent technologies, permanent magnet synchronous motor (PMSM) servo systems have seen increasing applications in industrial fields, accompanied by continuously rising control performance demands. Moreover, the adjustment of controller parameters is pivotal for the performance optimization of servo systems. This paper presents an optimization method for PMSM servo systems based on the coupling technique of the neural network surrogate model and intelligent optimization algorithm. A hybrid model is constructed by the proposed method, integrating a mathematical model based on transfer functions with an artificial neural network surrogate model, which is employed to compensate for the discrepancies between the mathematical model and the actual measured values. The accuracy and superiority of the hybrid model are comprehensively validated through training and validation loss analysis, fitting plot construction, and ablation experiments. Subsequently, based on the hybrid model, the qualitative and quantitative comparative analysis of the Pareto fronts of five commonly used multi-objective intelligent optimization algorithms is conducted. The optimal algorithm is determined through experimental validation of the optimization results to obtain the optimal result. The optimal result demonstrates that, compared to the initial result before optimization, the overshoot is reduced by 89.7%, and the settling time is reduced by 80.1%. Additionally, several other non-dominated solutions are available for selection, and all optimized results are superior to the initial result. This study provides a novel idea and method for the performance optimization of PMSM servo systems, contributing to the field with a robust and effective approach to enhance system control performance.

1. Introduction

Permanent magnet synchronous motor (PMSM) servo systems have garnered extensive applications across diverse fields, including automotive engineering, computer numerical control (CNC) machine tools, aerospace, and industrial robotics [1,2,3,4,5,6,7,8]. This widespread adoption is attributed to their stable operation, reliable structure, and high control precision. In recent years, the rapid advancement of intelligent manufacturing and unmanned operation technologies has imposed increasingly stringent demands on the control performance of PMSM servo systems [9,10,11]. Among the myriad controllers and control strategies that have been investigated and implemented, the proportional–integral–derivative (PID) controller continues to dominate industrial control applications. This prevalence is due to its robust versatility, straightforward control architecture, and remarkable flexibility. Nevertheless, the parameter tuning of PID controllers remains a pivotal challenge that warrants meticulous attention [12,13,14]. Conventional approaches to PID parameter tuning frequently depend on expert knowledge or manual trial-and-error adjustments by technicians, tailored to specific operational conditions. These methods, however, are not only time-consuming but also struggle to ensure the optimization and stability of control performance [15].

Nowadays, the rapid advancements in artificial intelligence and computational intelligence have significantly enhanced the application of intelligent optimization algorithms in PID controller parameter optimization. Prominent algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), gray wolf optimization (GWO), and evolutionary algorithm (EA) have gained widespread adoption [16,17,18,19,20]. These algorithms effectively navigate complex parameter spaces by emulating natural evolutionary processes or collective swarm behaviors, thereby addressing the inherent limitations of traditional methods that heavily depend on empirical knowledge and are susceptible to local optima. This paradigm shift not only improves the precision and efficiency of parameter tuning but also paves the way for more robust and adaptive control systems in various engineering applications.

On the other hand, the efficacy of intelligent optimization algorithms is highly contingent upon the accuracy of the system model being optimized. In practical engineering applications, the control performance of PMSM servo systems is influenced by a myriad of factors, including magnetic saturation effects, friction, motor losses, and the simplification or omission of certain subcomponents for computational convenience [21,22,23,24,25]. These complexities render the establishment of an accurate system model exceedingly challenging, thereby compromising the effectiveness and reliability of model-based optimization control strategies. In recent years, neural network surrogate models have emerged as effective tools for addressing complex system modeling challenges, owing to their data-driven nature, the advantage of not requiring consideration of internal system structural attributes, as well as their exceptional nonlinear mapping capabilities and robust generalization performance [26,27,28,29,30,31]. However, most current purely data-driven neural network surrogate models exhibit significant limitations: their fitting ability heavily relies on data quality, while errors and noise in real-world environments are inevitable; simultaneously, the lack of interpretability of black-box models also restricts their practical applications [32,33,34,35]. On the other hand, traditional mathematical models, although capable of accurately capturing physical reality, inevitably face challenges such as high modeling complexity and low computational efficiency when dealing with high-order nonlinear problems [36,37].

This study transcends the conventional “either-or” modeling approach by innovatively proposing a hybrid model that combines precision with residual learning capabilities. The model employs a transfer function mathematical model to capture the primary system dynamics and linear relationships, while utilizing an ANN to compensate for residual errors online. By integrating both components within a unified end-to-end training framework, the model retains the interpretability advantages of analytical models while gaining the nonlinear approximation capabilities of data-driven models. Furthermore, this study employs the proposed hybrid model as a high-fidelity surrogate model to systematically compare five mainstream intelligent optimization algorithms, aiming to minimize overshoot and settling time, thereby achieving global Pareto optimization of PID parameters.

The remainder of this paper is structured as follows. Section 2 offers a comprehensive overview of the PMSM servo system, elucidating its operational principles and key characteristics. Section 3 delves into the construction of the hybrid model, which integrates the mathematical model of the PMSM servo system, based on transfer functions, with an ANN surrogate model designed for compensation of the system. This section also delineates the optimization design variables, performance objective functions, and the procedural framework of the multi-objective optimization algorithm. Section 4 evaluates the prediction accuracy of the hybrid model and conducts a comparative analysis of five intelligent optimization algorithms in the context of PID parameter optimization. The optimal result is ultimately identified and validated through experimental results. Finally, Section 5 encapsulates the key findings and contributions of this study, providing a concise summary of the research outcomes.

2. PMSM Servo System

The PMSM servo system utilized in this study is based on a field-oriented control (FOC) strategy, specifically employing an $i_{dref}=0$ control method. The overall control structure is depicted in Figure 1. The system features a conventional dual-loop feedback control architecture, consisting of a speed control loop and a current control loop. The control process unfolds as follows. The reference speed is compared with the actual rotor speed obtained from motor feedback. The resulting speed error is then processed by the speed controller, which outputs the q-axis reference current $i_{qref}$ in the rotor rotating reference frame. In accordance with the $i_{dref}=0$ control strategy, the d-axis reference current $i_{dref}$ is set to zero. The error signals generated by comparing the d-axis and q-axis reference currents with the motor feedback currents serve as inputs to the current control loop. Following coordinate transformation, these signals are processed by a sinusoidal pulse width modulation (SPWM) module to generate PWM waves that drive the inverter circuit. The inverter then produces a series of pulse signals to simulate sinusoidal voltages, thereby energizing the motor. Both the speed and current loop controllers are typically implemented as PI controllers.

Figure 2 illustrates the measurement setup of the PMSM servo experimental platform, which consists of a power supply, an industrial control computer, a simulator, a driver, a PMSM, and a coaxial encoder. The industrial control computer is responsible for sending speed commands and adjusting the parameters of the PI controller, while the simulator is tasked with programming the programs into the driver. The driver integrates a dual-loop controller for current and speed, SPWM, and a three-phase inverter module. The encoder detects the rotor position of the motor in real time at a sampling frequency synchronized with the control cycle, converting the mechanical angle into electrical pulse signals that are fed back to the driver. Additionally, a first-order low-pass filtering algorithm is employed for signal processing to mitigate the impact of noise. The power supply for the measurement setup is sourced from Sichuan University in Chengdu, China, the motor is supplied by Panasonic Electric Works (China) Co., Ltd. in Chengdu, China, and the remaining components are sourced from Leetro Automation Technology Co., Ltd. in Chengdu, China.

3. Methodology

3.1. Construction of the Hybrid Model

3.1.1. Mathematical Model

Figure 3 and Figure 4 illustrate the transfer function block diagrams of the current loop and the speed loop, respectively, whose structures can be described by Equations (1) and (2).

The structure can be described by Equation (1):

$$G_c(s)={\displaystyle \frac{i_q(s)}{i_{qref}(s)}}={\displaystyle \frac{K_{pc}s+K_{ic}}{L_q{T}_1{s}^3+(L_q+R{T}_1)s^2+(R+K_{pc})s+K_{ic}}}$$

(1)

where $K_{pc}$ and $K_{ic}$ are the proportional and integral gains of the current loop PI controller, respectively; $K_{pwm}$ is the equivalent gain of the SPWM module; $T_s$ is the system control period; $T_1$ is the equivalent time constant of the system, which is set to 1.5 times $T_s$; $L_q$ is the q-axis inductance of the motor; $R$ is the stator resistance.

The structure can be described by Equation (2):

$$G_s(s)={\displaystyle \frac{N_r(s)}{N_{rcmd}(s)}}={\displaystyle \frac{45\varphi P_n{G}_c(s)(K_{ps}s+K_{is})(T_{d}s+1)}{45\varphi P_n{G}_c(s)(K_{ps}s+K_{is})+\pi s(T_{d}s+1)(T_{f}s+1)(Js+B)}}$$

(2)

where $K_{ps}$ and $K_{is}$ are the proportional and integral gains of the speed loop PI controller, respectively; $T_f$ is the time constant of the filter; $P_n$ is the number of pole pairs of the motor; $\varphi$ is the number of pole pairs of the motor; $J$ is the system moment of inertia; $B$ is the damping coefficient; $T_d$ is the time delay introduced by speed detection, which is set to 0.5 times $T_s$.

In practical systems, the SPWM module can be approximated as a linear element with a gain of 1, while the inverter can be modeled as a delay element with a time delay of 0.5 control periods. Additionally, the digital controller introduces a further delay of one control period. To simplify the analysis, these delay elements are typically approximated as first-order inertial components whose time constant is set to 1.5 control periods. Under the $i_{dref}=0$ control strategy, a voltage feedforward compensation method is employed to effectively eliminate the influence of the motor speed, flux linkage dynamics, and d-axis inductance coupling terms on the q-axis voltage equation, thereby simplifying the controller optimization. The first-order low-pass filter can be approximated as a first-order inertial element. Similarly, the delay introduced by the encoder during position detection can also be regarded as an inertial element in the feedback path. These components affect the system’s phase margin and response speed to a certain extent and, therefore, must be carefully considered during controller optimization.

3.1.2. Artificial Neural Network Surrogate Model

The ANN is a computational model designed to emulate the structure and functionality of biological neural networks. It consists of a vast number of interconnected neurons (nodes), each linked through connection weights. Neurons serve as the fundamental computational units of the network: they receive input signals, perform weighted summation, introduce nonlinearity via an activation function, and subsequently produce an output. ANNs are typically structured into three main components: an input layer, one or more hidden layers, and an output layer. The depth of the network is determined by the number of hidden layers, which play a crucial role in enabling the network to learn and represent complex patterns and relationships in the data.

To enhance the fitting accuracy of the neural network, two deep neural networks with two inputs and one output were constructed. The inputs are the design variables corresponding to the parameters of the speed loop PI controller, and the outputs are the differences between the overshoot and settling time calculated by the PMSM servo system model and the measured values. The training of the neural network is implemented using Python software version 3.13. To determine the appropriate network architecture, this paper takes the structure from similar tasks in the literature as an initial anchor, constructs a full factorial grid of 2, 3, and 4 hidden layers with 8, 16, 32, and 64 neurons per layer, and conducts a systematic search using the training–validation loss as the evaluation metric [38,39,40]. The structure of the deep neural network is shown in Figure 5, which includes four fully connected hidden layers with 64, 32, 8, and 8 neurons, respectively. The activation function used is the LeakyReLU function with a hyperparameter of 0.2.

In this study, the optimal Latin hypercube sampling method was employed to thoroughly search the design space, resulting in a dataset of 1600 samples. The dataset was divided into training, validation, and testing sets in a ratio of 7:1:2 to set appropriate hyperparameters. The neural network was trained using the Adam optimizer. The accuracy of the neural network surrogate model was compared by calculating evaluation metrics. The commonly used regression model evaluation metrics include the coefficient of determination ($R^2$), root mean square error (RMSE), mean absolute error (MAE), and mean squared error (MSE). The calculations of these four regression evaluation metrics can be expressed as follows:

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(y_i-\stackrel{\wedge }{y_i}}{)}^2}{{\displaystyle \sum _{i=1}^n(y_i-\stackrel{-}{y}}{)}^2}}$$

(3)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(y_i-\stackrel{\wedge }{y_i}}{)}^2}$$

(4)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-\stackrel{\wedge }{y_i}\right|}$$

(5)

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(y_i-\stackrel{\wedge }{y_i}}{)}^2$$

(6)

where $y_i$ is the true value; $\stackrel{\wedge }{y_i}$ is the predicted value; $n$ is the number of samples; and $\stackrel{-}{y}$ is the mean of true values.

3.2. Optimization of PMSM Servo Systems

3.2.1. System Optimization Parameters and Performance Indicators

In this study, the response speed of the servo system to a step signal is primarily evaluated using two performance metrics: overshoot and settling time. The two metrics collectively constitute the objective function of the optimization problem, where the unit of overshoot is percentage (%) and the unit of settling time is second (s). Overshoot is defined as the ratio of the maximum deviation in the system output beyond its steady-state value to the steady-state value itself during a step response. A smaller overshoot indicates a smoother dynamic response and helps avoid excessive speed fluctuations. Settling time is defined as the time required for the system output to enter and remain within a specified tolerance band around the steady-state value following a step input. A shorter settling time reflects a faster response and better dynamic performance.

In practical engineering applications, due to the significant order-of-magnitude difference in bandwidth between the current loop and the speed loop, even minor perturbations in the current loop gains can lead to severe oscillations in the speed output. To avoid system oscillations and achieve favorable dynamic response, the PI parameters of the current loop in this study are fixed to a set of empirically tuned values, while the optimization efforts are primarily focused on the PI controller parameters of the speed loop. Furthermore, due to the relatively small magnitude of the speed loop PI parameters and the significant difference in scale between the proportional and integral gains, direct optimization of the original parameters may lead to inefficient search behavior. To address this issue, a set of new design variables is introduced, which are mapped to the actual PI parameters through a transformation function. This approach enhances numerical stability and convergence efficiency during the optimization process. The mapping relationship is expressed as Equations (7) and (8):

$$K_p={\displaystyle \frac{J{I}_r{K}_{ps}\lambda (1+\tau )}{T_r}}$$

(7)

$$K_i={\displaystyle \frac{\delta K_p{T}_s}{K_{is}}}$$

(8)

where $K_p$ and $K_i$ are the design variables to be optimized; $\tau$ is the inertia ratio; $I_r$ is the rated current; $T_r$ is the rated torque; and $\lambda$ and $\delta$ are constants.

The main parameters involved in the system and their corresponding values are listed in

Table 1. Main parameters and values of the PMSM servo system.

Parameter	Value
Design variable $K_p$	[75, 900]
Design variable $K_i$	[80, 800]
Stator resistance $R$	1.87 Ω
q-axis inductance $L_q$	3.4 × 10−3 mH
Number of pole pairs $P_n$	5
Magnetic flux linkage $\varphi$	6.78 × 10−2 Wb
Magnetic flux linkage $J$	5.2 × 10−5 kg·m2
Damping coefficient $B$	0 N·m·s/rad
System control period $T_s$	1/12,000 s
Filter time constant $T_f$	8.4 × 10−4 s
Inertia ratio $\tau$	0.75
Rated current $I_r$	2.5 A
Rated torque $T_r$	1.27 N·m
System constant $\lambda$	6.5797362 × 10−2
System constant $\delta$	1000

.

3.2.2. Multi-Objective Optimization Algorithms

The multi-objective optimization problem refers to the task of optimizing multiple objective functions simultaneously while satisfying a set of constraints. Typically, these objective functions are in conflict with each other, making it difficult to find a solution that satisfies all objectives at the same time. In this study, the two objective functions of the PMSM servo system optimization problem, namely overshoot and settling time, are mutually exclusive. A larger overshoot indicates a faster response of the system to a step signal, but it also leads to an increased settling time for the system to converge to stability and may even induce system oscillation. The Pareto optimization model is one of the most commonly used models in multi-objective optimization. A Pareto optimal solution is defined as a solution where it is impossible to improve one objective without deteriorating at least one other objective. Although the Pareto model cannot directly provide the optimal solution to the optimization problem, it can eliminate inferior solutions and obtain a widely distributed and uniformly spaced set of Pareto optimal solutions, allowing decision-makers to select the most suitable solution based on practical needs.

Intelligent optimization algorithms have been proven to be effective methods for obtaining the Pareto optimal solution set for multi-objective optimization problems. These algorithms approximate the optimal solution set of multi-objective problems by simulating biological evolution or swarm behavior in nature. In this study, five commonly used intelligent optimization algorithms, namely NSGA-II, MOGWO, SPEA2, MOEA/D, and MOPSO, are employed to solve the PMSM servo system optimization problem and obtain the optimal solution set. The five optimal solution sets are compared and analyzed to select the best design. Among them, the population size for the five optimization algorithms was set to 100, and the maximum number of iterations was set to 100. The crossover and mutation probabilities were set to 1 and 0.1, respectively. NSGA-II is a multi-objective optimization algorithm based on genetic algorithms. It employs fast non-dominated sorting, the elitist-preserving strategy, and the crowded-comparison operator to achieve efficient global search and maintenance of solution diversity in multi-objective optimization problems, as shown in Figure 6.

Convergence, distribution, and optimization time are three crucial metrics for evaluating multi-objective optimization algorithms. Various indicators can be employed to assess the performance of optimization algorithms. Commonly used quantitative methods for evaluating optimizers include the generational distance (GD) for assessing solution set convergence, the spacing metric based on distance (Spacing) for evaluating solution set distribution, and the hypervolume (HV) and inverted generational distance (IGD) for comprehensive evaluation [41,42,43,44]. In this study, HV and IGD are selected as performance metrics to evaluate the convergence, accuracy, and diversity of the Pareto front, while optimization time is used to assess the efficiency of the algorithm. This approach helps identify the best-performing optimizer among the five optimization algorithms considered. All Pareto front solutions from independent run cycles of the five optimization algorithms are combined. Non-domination and duplication checks are applied to this augmented Pareto front set to filter it and create the “true Pareto front.” The HV, IGD, and optimization time values of the obtained Pareto front from each optimization algorithm relative to this simulated “true Pareto front” are calculated to evaluate the performance of each optimizer. HV and IGD can be computed using Equations (9) and (10), respectively.

$$IGD(P,P^{*})={\displaystyle \frac{1}{\left|P^{*}\right|}}{\displaystyle \sum _{p^{*}\in P^{*}}{\underset{p\in P}{{\displaystyle min}}}^{d(p,p^{*})}}$$

(9)

$$HV(P,r)=Volume(\underset{p\in P}{U}[p_1,r_1]\times [p_2,r_2]\times \cdot \cdot \cdot \times [p_m,r_m])$$

(10)

where $P$ and $P^{*}$ represent the set of points in the currently obtained Pareto front and the simulated true Pareto front, respectively. $p$ and $p^{*}$ denote individual points in $P$ and $P^{*}$, $d(p,p^{*})$ represents the Euclidean distance between two points, $m$ is the number of objective functions, $r$ is a set of reference points, and $[p_m,r_m]$ denotes the interval from $p_m$ to $r_m$ on the $m$-th objective. Larger HV values, smaller IGD values, and shorter optimization times reflect better performance and efficiency of the Pareto front.

4. Results

4.1. Model Evaluation Results

Figure 7a,b depict the loss curves of the two neural networks on the training and validation sets, which are employed to predict the discrepancies between the calculated values from the mathematical models and the actual measured values of overshoot and settling time in the servo system. The loss function employed is the MSE. Prior to training, the labels of both the training and validation sets were normalized. As the number of iterations increased, both the training and validation losses decreased rapidly and eventually stabilized. Moreover, the minimal difference between the training and validation losses indicates that the constructed models possess high prediction accuracy and good generalization capability.

As illustrated in Figure 8, (a) and (b) display the scatter plots of the predicted values versus the measured values for the two optimization objectives, namely overshoot and settling time, as well as their respective coefficients of determination, which are 0.9949 and 0.9807. The horizontal axis represents the actual measured values of overshoot and settling time in the servo system, while the vertical axis denotes the predicted values obtained from the hybrid model combining the artificial neural network surrogate model and the mathematical model developed in this study. The red dashed line indicates the fit where the predicted values are equal to the measured values, and the points closer to this line represent higher prediction accuracy. It can be clearly observed from the figure that the hybrid model exhibits excellent linear fitting for the actual measured values of both objective functions, thereby demonstrating the high reliability of the artificial neural network surrogate model and the hybrid model.

4.2. Ablation Experiments

Figure 9a,b present the fitting plots of the overshoot and settling time calculated by the mathematical model compared with the measured values. The results indicate that the calculated overshoot values from the mathematical model are generally higher than the measured values, while the calculated settling time values are consistently lower than the measured ones. This discrepancy increases with the magnitude of the values. It is speculated that this deviation arises because the servo system is influenced by factors such as motor magnetic saturation, friction, and losses, which exhibit complex nonlinear characteristics. The ideal model constructed by the mathematical model fails to adequately account for these factors.

Table 2. Performance metrics of the hybrid model and other models.

Models	Overshoot				Settling Time
	RMSE	MSE	MAE	R2	RMSE	MSE	MAE	R2
Hybrid	$0.391$	$0.153$	$0.293$	$0.995$	$3.51\times {10}^{-3}$	$1.23\times {10}^{-5}$	$1.87\times {10}^{-3}$	$0.981$
Surrogate	$0.485$	$0.236$	$0.381$	$0.992$	$4.73\times {10}^{-3}$	$2.24\times {10}^{-5}$	$2.40\times {10}^{-3}$	$0.965$

lists the evaluation metrics used to assess the prediction accuracy of the two regression models, while Figure 10 illustrates the Pareto fronts obtained by the two models using the NSGA-II optimization algorithm, with the legend indicating the optimization time. The hybrid model integrates a neural network with a mathematical model to achieve prediction functionality, where the neural network component specifically predicts the error between the model-calculated values and the measured values of the servo system’s overshoot and settling time. In contrast, the surrogate model directly predicts the actual measured values of the servo system’s overshoot and settling time using an ANN model. As shown in Table 2, both models exhibit high prediction accuracy for the system’s actual measured values, thereby validating the reliability of the artificial neural network surrogate model and the hybrid model constructed in this study. Further comparison of the four evaluation metrics and the Pareto fronts in Figure 10 reveals that the hybrid model outperforms the surrogate model in all metrics, and the Pareto front obtained based on the hybrid model largely dominates that obtained based on the surrogate model. This indicates that the hybrid model has a significant advantage in fitting the nonlinear servo system, especially in resource-constrained scenarios for surrogate-assisted optimization. Under the same algorithmic framework, the optimization time of the hybrid model increases compared to the pure surrogate model. This phenomenon primarily stems from the computational complexity of the mathematical model component within the hybrid model.

In engineering, the settling time typically refers to the duration from when the tracking speed first enters the range of $\pm 2\%$ of the steady-state value and thereafter remains within this boundary without exceeding it. In experimental measurements, the settling time is determined by the discrete sampling point that first satisfies this condition, with the corresponding time recorded as the settling time. As shown in the validation results in

Table 3. Validation results of the optimal solutions for different models.

Result Index	Hybrid Model		Surrogate Model		Mathematical Model
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
1	0.281	0.0160	0.342	0.0117	0.708	0.0089
2	0.305	0.0153	0.525	0.0115	0.839	0.0088
3	0.342	0.0117	0.647	0.0083	0.891	0.0081
4	0.525	0.0078	0.656	0.0079	1.127	0.0080
5	0.708	0.0077	0.708	0.0077
6	0.787	0.0076	0.787	0.0076
7	0.983	0.0075	0.983	0.0075

, the optimal solutions of the hybrid model and the surrogate model exhibit a uniform distribution in terms of the overshoot objective function, while displaying a certain step-like span in the settling time. This trend is consistent with the Pareto front shown in Figure 10, and the optimal solution set of the hybrid model predominantly dominates those of the surrogate model and the mathematical model, demonstrating superior performance in terms of distribution, particularly within the range of small overshoot objectives. In Figure 11, the vertical axis represents the tracking speed of the motor, while the horizontal axis denotes the control step number (i.e., the number of control cycles experienced). By comparison, it can be observed that the optimal solution of the hybrid model has the largest distribution range of overshoot, followed by the surrogate model. Within the same model, a faster rise in tracking speed implies a shorter time for the system to reach stability, but also a correspondingly larger overshoot, which is consistent with the validation results of the same model in Table 3. Result 2 in Figure 11 may be attributed to random system errors.

4.3. Comparison of Algorithms and Optimal Results

Figure 12 visually compares the convergence and distribution of the Pareto fronts for the five optimization algorithms, with the legend indicating the optimization time of each algorithm.

Table 4. Three performance and efficiency evaluation metrics for the five algorithms.

Algorithms	IGD	HV	Optimization Time
MOEA/D	1.408	0.659	353.2
MOPSO	0.212	0.992	78.8
SPEA2	0.306	0.904	62.3
MOGWO	0.208	0.992	434.1
NSGAII	0.206	0.992	64.3

lists the comprehensive performance and efficiency evaluation metrics for these algorithms. In Figure 12, the Pareto front of MOEA/D exhibits poor distribution and significantly longer optimization time. The Pareto fronts of NSGA-II, MOGWO, and MOPSO demonstrate similar convergence, with their distribution outperforming that of SPEA2. However, the optimization times of NSGA-II and SPEA2 are nearly identical, followed by MOPSO, while the optimization time of MOGWO far exceeds that of the other algorithms, aligning with the numerical results in Table 4. Consequently, the ranking of the five algorithms is as follows: NSGA-II > MOPSO, SPEA2 > MOGWO > MOEA/D.

Table 5. Validation results of the optimal solutions for different algorithms.

Result Index	NSGA-II		SPEA2		MOPSO
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
1	0.281	0.0160	0.311	0.0140	0.250	0.0152
2	0.305	0.0153	0.342	0.0139	0.342	0.0138
3	0.342	0.0138	0.491	0.0084	0.403	0.0126
4	0.525	0.0079	0.525	0.0079	0.525	0.0080
5	0.598	0.0078	0.617	0.0078	0.671	0.0079
6	0.708	0.0077	0.708	0.0077	0.708	0.0078
7	0.787	0.0076

and Figure 13 present the validation results of the Pareto fronts for the three algorithms, namely NSGA-II, SPEA2, and MOPSO, as well as the corresponding tracking curves. Overall, NSGA-II significantly outperforms MOPSO and SPEA2 in terms of search range and the number of optimal solutions. MOPSO performs better than SPEA2 in the search range of small overshoot and long settling time. However, in terms of the quality of optimal solutions, the optimal solution sets of the three algorithms are non-dominated when random error disturbances are ignored. This conclusion is consistent with the comparison of the Pareto fronts of the three algorithms shown in Figure 12. Moreover, Figure 13 clearly demonstrates that the solution search ranges of NSGA-II and MOPSO are superior to that of SPEA2. For the same algorithm, the optimal solution with a larger overshoot has a faster rise in tracking speed, and consequently, a shorter settling time.

Table 6. Optimization results.

	${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	Overshoot	Settling Time
Initial	270	210	5.10	0.0392
Optimal	428	462	0.525	0.0078

and Figure 14 present detailed information on the optimization results and the corresponding tracking curves, respectively. The solution with an overshoot of 0.525% and a settling time of 0.0078 s was ultimately selected as the optimal solution. This is because further optimization of the overshoot would lead to a significant increase in the settling time, while optimization of the settling time would result in a substantial increase in the overshoot with only a small change. Compared with the results before optimization, the increase in the design variable $K_p$, i.e., the proportional coefficient in the speed loop PI controller, leads to a reduction in the settling time while increasing the overshoot. The increase in the design variable $K_i$, i.e., the reduction in the integral coefficient in the speed loop PI controller, indirectly reduces the overshoot while increasing the settling time. The optimal solution obtained in this study achieved an 89.7% reduction in the overshoot optimization target and an 80.1% reduction in the settling time optimization target. As shown in Figure 14, compared with the tracking curve before optimization, the optimized result has a smaller peak, i.e., a smaller overshoot, and the system can quickly reach stability at a control step number of 100, corresponding to a settling time of approximately 0.0083 s. Therefore, the proposed method of coupling the neural network surrogate model with the intelligent optimization algorithm, as presented in this paper, can be proven to be an effective approach for optimizing the PMSM servo system.

5. Conclusions

The increasing demand for high control performance in PMSM servo systems, which are widely used in industrial applications, has made their performance optimization a critical issue. This study applies the technique of coupling neural network surrogate models with intelligent optimization algorithms to the optimization of PMSM servo system control performance. A hybrid model combining mathematical models and ANN surrogate models was established, and its superiority was demonstrated through accuracy verification and ablation experiments. Furthermore, based on the hybrid model, the performance of five commonly used multi-objective intelligent optimization algorithms was tested, and the best-performing algorithm was identified through the analysis of the Pareto front and three quantitative evaluation metrics. The three significant results obtained during this study are as follows:

(1)

A hybrid model for the PMSM servo system that more accurately approximates the actual measured values has been successfully constructed. This model integrates a mathematical model based on transfer functions and an ANN surrogate model used to fit the discrepancies between the calculated values from the mathematical model and the actual measured values. The training and validation losses for the two optimization objectives of the hybrid model are close in value and consistent in trend, with the coefficients of determination for the fits between the measured and predicted values of the two optimization objectives being 0.995 and 0.981, respectively. Based on evaluation metrics such as RMSE and R², as well as experimental validation results of the optimal solution sets for the three models, the models are ranked in terms of precision as follows: hybrid model > surrogate model > mathematical model.

(2)

A comparative analysis of the Pareto fronts and quantitative performance evaluation metrics for five commonly used multi-objective optimization algorithms, namely NSGA-II, MOEA/D, MOPSO, MOGWO, and SPEA2, was conducted. The optimal solution sets of the three best-performing algorithms were experimentally validated, and their tracking curves were compared. The performance ranking of the five algorithms is as follows: NSGA-II > MOPSO, SPEA2 > MOGWO > MOEA/D.

(3)

The coupling technique of neural network surrogate models with intelligent optimization algorithms has been validated as an effective method for optimizing the response speed of PMSM servo systems. Compared with the results before optimization, the selected optimized results achieved an 89.7% reduction in the overshoot optimization target and an 80.1% reduction in the settling time optimization target. Moreover, several non-dominated solutions are available for selection, and all optimized results are superior to those before optimization.