Review of Intelligent Motor Controller Parameter Self-Tuning Technology

Abstract

The self-tuning of control parameters for permanent magnet synchronous motor controllers is extensively utilized in industrial and production settings. The self-tuning algorithms and strategies employed significantly influence the quality and efficiency of production processes. In response to diverse practical application scenarios and system performance requirements, scholars have developed numerous intelligent self-tuning schemes. This paper reviews a substantial body of recent research on intelligent self-tuning technologies for permanent magnet synchronous motor controller parameters conducted by international scholars. It summarizes typical intelligent self-tuning methods, including single-neuron proportional–integral–derivative controllers, neural network proportional–integral–derivative controllers, and proportional–integral–derivative controllers optimized using particle swarm optimized algorithms, and compares their performance metrics through simulation studies. Additionally, it outlines the self-tuning strategies and optimization improvements based on each intelligent algorithm, identifies key research challenges, and evaluates existing solutions. Finally, this paper provides an overview of the current state and future prospects of intelligent self-tuning technology for permanent magnet synchronous motor controller parameters.

1. Introduction

Permanent magnet synchronous motors (PMSMs) have gained extensive application in motion control systems, including industrial robots and machine tools, owing to their superior power density, high torque-to-inertia ratio, and broad speed control range [1]. Proportional–integral–derivative (PID) controllers, characterized by their straightforward structure, ease of implementation, and robust performance, are predominantly employed in PMSM drive systems [2,3]. The improper setting of PID controller parameters can significantly impact system performance. Consequently, researching the tuning methods for PID controller parameters in PMSM drive systems holds substantial engineering significance [4]. The control parameter tuning methods can primarily be categorized into two types: off-line tuning and on-line tuning. Off-line parameter tuning typically involves manual adjustments based on system response frequency domain indicators. On the other hand, on-line parameter tuning methods encompass several approaches, including rule-based control parameter self-tuning, model reference adaptive control parameter self-tuning, extremum tuning, active disturbance rejection control in combination with self-tuning, and intelligent algorithm-based control parameter self-tuning. The intelligent algorithm-based self-tuning method primarily adjusts the control parameters using artificial intelligence algorithms and numerical optimization techniques. The classification of tuning methods for the motor PID controller is illustrated in Figure 1.

The method for off-line tuning PID control parameters based on frequency domain characteristic analysis has reached a relatively mature stage. Scholars have conducted a series of control parameter tuning investigations based on the frequency domain response characteristics of the system, such as amplitude margin, phase margin, and the distribution of closed-loop poles and zeros. For instance, in Reference [5], a physical PID controller prototype with an automatic tuning function was designed by combining the Marchetti-Scali method and the Ackermann method for calculating PID control parameters, thereby significantly enhancing the effectiveness of PID parameter tuning via the relay method. Reference [6] investigated the impact of the dominant pole and phase margin on system performance under various PID control parameters. In industrial application scenarios, due to the nonlinear and time-varying uncertainties inherent in control systems, the off-line tuning method for PID control parameters based on frequency domain characteristic analysis often fails to meet the required performance standards.

The development of online control parameter self-tuning methods has been rapid. Numerous scholars have employed heuristic algorithms or fuzzy logic to determine the optimal values of control parameters, ensuring the efficiency and responsiveness of PID controllers. This is typically achieved through in-loop simulations using either the built-in PID controller of a programmable logic controller (PLC) or an external parameter self-tuning algorithm module. Reference [7] established communication between a PLC and MATLAB via the OPC (OLE for process control) UA (unified architecture), proposing a PLC-based PID controller adjustment method. The study further improved the self-tuning process of control parameters by adopting an adaptive artificial bee colony fuzzy logic (AABC-FL) approach. Reference [8] utilized MATLAB/Simulink to design a PSO (particle swarm optimized) fuzzy PID controller based on PLCs, significantly enhancing the effectiveness and real-time applicability of the system in industrial frequency control.

Given that the self-tuning of PID control parameters based on PLC has disadvantages, such as stringent configuration requirements and susceptibility to limitations imposed by initial conditions, Reference [9] proposed a PID controller integrating extremum search control and feedforward compensation, which effectively reduced energy consumption during system operation. Additionally, some scholars have combined model reference adaptive control, active disturbance rejection control, and PID control to further enhance the control performance [10,11].

In recent years, the advancement of intelligent algorithms, including single-neuron models, neural networks, and genetic algorithm (GA), has demonstrated significant advantages in controlling nonlinear, time-varying, and other complex systems. Researchers have integrated these intelligent algorithms with traditional PID control strategies to achieve intelligent self-tuning of PID parameters. Consequently, various intelligent self-tuning methods for motor controller parameters have been proposed, such as those based on single-neuron PID controllers, neural network PID controllers [12], and genetically optimized PID controllers [13].

An intelligent algorithm does not require an exact mathematical model of the control system. Instead, it evaluates system states using performance metrics such as time series integral error [14] and cross entropy [15], iteratively optimizing control parameters to enhance the relevant performance indices of the control system. The block diagram illustrating the self-tuning of control parameters based on the intelligent algorithm is presented in Figure 2.

Focusing on the intelligent self-tuning method of motor controller parameters, this paper elucidates the current advancements in utilizing intelligent algorithms for control parameter self-tuning. Using a simulation, it compares the performance characteristics of three types of intelligent self-tuning technologies for motor controller parameters, summarizes the primary research challenges and existing solutions, and concludes with an overview and future outlook.

2. Single-Neuron PID Controller

Since the 1980s, the integration of artificial neural networks (ANNs) with control systems has emerged as a novel branch of intelligent control, providing an innovative approach to addressing the challenges posed by complex, nonlinear, uncertain, and time-varying systems [16].

The single neuron serves as the fundamental building block of neural networks, possessing self-learning and adaptive capabilities. Its simplicity and computational efficiency make it well-suited for the self-tuning of proportional control system parameters [17].

In Figure 3, the controller output at the kth sampling time can be mathematically represented as follows:

$$x_1(k)=e(k)-e(k-1)$$

(1)

$$x_2(k)=e(k)$$

(2)

$$x_3(k)=e(k)-2e(k-1)+e(k-2)$$

(3)

$$u(k)=u(k-1)+w_1{x}_1(k)+w_2{x}_2(k)+w_3{x}_3(k)$$

(4)

${\eta }_p, {\eta }_i{, \mathrm{a}\mathrm{n}\mathrm{d} \eta }_d$ represent the proportional, integral, and derivative iterative adjustment coefficients, respectively. The iterative adjustment formula for the neuron weights is presented as follows.

$$w_1(k)=w_1(k-1)+{\eta }_p{x}_1(k-1)u(k-1)e(k-1)$$

(5)

$$w_2(k)=w_2(k-1)+{\eta }_i{x}_2(k-1)u(k-1)e(k-1)$$

(6)

$$w_3(k)=w_3(k-1)+{\eta }_d{x}_3(k-1)u(k-1)e(k-1)$$

(7)

As illustrated in Figure 3, $e$ represents the deviation between the actual system output and the desired input, u denotes the output from the embedded PID controller, k signifies the proportional gain of the PID controller, $x$₁ corresponds to the proportional error, $x$₂ to the integral error, and $x$₃ to the derivative error of the embedded PID controller. The neuron weights $w$₁, $w$₂, and $w$₃ are utilized as the proportional, integral, and derivative gains of the PID controller, respectively, enabling the realization of proportional control parameter self-tuning through the iterative adjustment of these weights.

Scholars have conducted extensive research on the self-tuning technology of control parameters for single-neuron PID controllers. Firstly, regarding improvements to the neuron weight iteration formula in control parameter self-tuning, reference [16] introduced a weighted sum of squares approach to enhance the iteration formula and modified the gain coefficient K into a dynamic parameter, thereby improving the efficiency of single-neuron control parameter self-tuning. Reference [18] analyzed the performance differences in control parameter self-tuning under various iterative learning rates of neuron weights through simulations and proposed a method for determining the optimal learning rate. Reference [19] developed a quadratic performance exponential learning algorithm for the iteration of single-neuron weights. This approach exhibits superior adaptability and anti-interference capabilities, effectively reducing system overshoot. To further enhance the frequency response capability of self-tuning control parameters, reference [20] utilized GA to optimize the weight iteration formula of the single neuron, improving system robustness.

Secondly, advanced control methods have been employed to directly improve the control structure of the single-neuron PID controller. Liu Chunyuan et al. from Qingdao University of China proposed a compound control parameter self-tuning method. When motor speed is below 2000 rpm, a single-neuron PID controller is used; when the speed exceeds 2000 rpm, a traditional PID controller is employed to improve the response speed of the control system [17]. Since the single-neuron PID controller is not sensitive to angle deviation signals, Ma Fei et al. from Xi’an Jiaotong University of China established an angle-predetermined loop model to enhance the self-tuning link of control parameters and improve the tracking accuracy of the control system for angle-given signals [21]. To address the poor multi-objective optimization performance of the single-neuron control parameter self-tuning system, Jiao Jun’s research team at Hefei University of Technology used a radial basis function neural network (RBFNN) to assess the system state. This method leverages the rapidity of single-neuron self-tuning while avoiding local optimal solutions [22].

A significant drawback of single-neuron control parameter self-tuning is its limitation to setting only the proportional control parameters of the PID controller, without real-time adjustment of integral and derivative control parameters. Zhang Hui et al. from Anhui University of China employed Hebbian computing strategies to adjust proportional control parameters in real-time, ensuring system performance comparable to the simultaneous tuning of integral and derivative control parameters [23]. The initial value setting of neuron weights significantly impacts the efficiency of self-tuning. Xu Song et al. from Jiangsu University of China adopted the adaptive moment estimation (Adam) method to set the initial value of single-neuron weights, accelerating the self-tuning process [24]. To address the poor self-tuning effect of single-neuron PID control parameters in strongly coupled and complex nonlinear systems, Cao Hongliang et al. from Huazhong University of Science and Technology combined feedforward control structures with single-neuron PID controllers, resulting in a system that combines robust control and PID control advantages [25].

The single-neuron PID control parameter self-tuning system, despite having online tuning capabilities for PID control parameters, suffers from limited nonlinear approximation and logic operation abilities due to its single-layer neural network structure. Consequently, it often fails to achieve satisfactory control effects in complex systems [26].

In summary, scholars accelerated the self-tuning speed of control parameters by enhancing the iterative approach for single-neuron weight adjustment. These improved methods exhibit advantages such as a simple structure and ease of implementation. Additionally, researchers have enhanced the control architecture of the single-neuron PID controller. This effectively addresses the limitation of weak multi-objective optimization capability in the single-neuron PID controller, resulting in a more flexible control framework. The improved methods and their corresponding advantages for the single-neuron PID controller are summarized in

Table 1. Enhanced methods and corresponding advantages of single-neuron PID controllers.

Self-Tuning Method for Control Parameters	Enhanced Methods	Key Features
Single-neuron PID controller	Sum of squares in weighting calculation	Enhanced efficiency
	Optimization of fixed learning rate	Enhanced self-tuning property
	Quadratic performance-based exponential learning algorithm	Enhanced adaptability and anti-jamming capabilities
	Optimization of GA	Enhanced robustness
	Compound control model	Avoid local optimal solution
	RBFNN state identification	Improved control performance
	Hebbian computational methodology	Self-tuning speed increased
	Stochastic optimization of adaptive momentum	Withstand coupling and nonlinearity
	Implement of feedforward control structure	Combination of robust, PID control

.

3. Neural-Network-Based PID Controller

Neural networks have the capability to learn and store a large number of input–output mappings that accurately reflect the underlying system model without requiring an explicit derivation of the detailed mathematical representation of complex systems [27]. The structure is illustrated in Figure 4. In Figure 4, $e\left(k\right), e\left(k-1\right) \mathrm{a}\mathrm{n}\mathrm{d} e\left(k-2\right)$ represent the deviations between the actual system output and the desired input at the kth, (k − 1)th, and (k − 2)th sampling times, respectively. The weights of the neural network are denoted as $w_{ij}$, while a and b represent the bias constants of the neural network. The outputs of the neural network serve as the control parameters for the PID controller. If the activation function of the input layer is $f\left(x\right)$ and the normalization function of the output layer is $g\left(x\right)$, the network weights $w_{ij}$ are updated iteratively, and the output $O_i^3$ of the neural network can be expressed as follows:

$$I_1^{(1)}=e(k)$$

(8)

$$I_2^{(1)}=e(k-1)$$

(9)

$$I_3^{(1)}=e(k-2)$$

(10)

$$O_1^{(3)}=K_p$$

(11)

$$O_2^{(3)}=K_i$$

(12)

$$O_3^{(3)}=K_d$$

(13)

$$O_i^{(3)}=g\left({\displaystyle \sum _i{w}_{li}^{(2,3)}f}\left({\displaystyle \sum _j{w}_{ij}^{(1,2)}{I}_i^{(1)}+a}\right)+b\right)$$

(14)

Let ${\eta }_1 \mathrm{a}\mathrm{n}\mathrm{d} {\eta }_2$ denote the iterative learning rates. ${\alpha }_1 \mathrm{a}\mathrm{n}\mathrm{d} {\alpha }_2$ represent the iterative inertia coefficients. The weight update rule for the neural network can then be formulated as follows:

$$w_{ij}^{(1,2)}(k)={\eta }_1{O}_j^1(k)+{\alpha }_1{w}_{ij}^{(1,2)}(k-1)$$

(15)

$$w_{li}^{(2,3)}(k)={\eta }_2{O}_i^2(k)+{\alpha }_2{w}_{li}^{(2,3)}(k-1)$$

(16)

Scholars have conducted extensive research on the self-tuning technology of neural network PID control parameters. Firstly, optimization efforts have focused on refining the iterative process of neural network weights and the self-tuning calculation of control parameters. Reference [28], in response to the inherent challenges of initializing weights and thresholds in BP neural networks, utilized the artificial fish swarm algorithm (AFSA) to optimize the selection of initial weights in this study. Consequently, they designed the AFSA-BP-PID controller, which significantly improved the dynamic performance of the permanent magnet synchronous linear motor and reduced the system tuning time. Reference [29] enhanced the neural adaptive iterative learning control scheme by incorporating an error tracking signal, achieving satisfactory non-uniform angle trajectory tracking performance. Figure 5 illustrates the enhanced block diagram of the control parameter self-tuning mechanism achieved through the integration of feedforward control.

Abdullah Fayez et al. from the University of Philadelphia, Jordan, compared the performance differences between PID control parameter self-tuning systems optimized using fuzzy logic and those optimized by neural networks through simulations, proposing a formula to determine the optimal learning rate for network weight iteration [30]. Arshia Rezaei’s team at Sharif University of Technology in Iran introduced the cuckoo search algorithm to design a nonlinear model predictive control (MPC) algorithm with neural networks for self-tuning, enhancing system robustness [31]. Figure 6 presents the block diagram illustrating the structure of the neural network PID controller within the nonlinear model predictive control algorithm.

To conserve computing resources and enhance the efficiency of control parameter self-tuning, reference [32] employed a diagonal recursive optimization algorithm to streamline the computational steps of the output layer in the neural network PID control parameter self-tuning method, thereby accelerating the convergence speed of the control parameter self-tuning process. Figure 7 presents the block diagram of the neural network PID controller structure, which has been simplified through the application of the diagonal recursive optimization algorithm.

Secondly, various research teams have applied reinforcement learning and other techniques to directly enhance the structure of neural network PID controllers, effectively boosting system performance. To mitigate the impact of model and sensor measurement errors on control parameter self-tuning, reference [33] designed an adaptive compensator based on Lyapunov theory, improving the robustness of neural network PID controllers.

Traditional backpropagation neural network (BPNN) PID control parameter self-tuning methods suffer from long iteration cycles and slow response times. Reference [34] improved this traditional method using ANN, significantly enhancing the rapidity of self-tuning. Figure 8 presents the block diagram illustrating the structure of the ANN-PID controller.

Reference [35] combined MPC with neural network PID control parameter self-tuning, improving dynamic performance. Despite these improvements, MPC-optimized neural network PID controllers still struggle with nonlinear interference suppression. Reference [36] further refined the neural network PID control parameter self-tuning method using actor–critic reinforcement learning, balancing flexibility and stability. Figure 9 presents the structural block diagram of the neural network PID controller enhanced by the Actor-Critic reinforcement learning approach.

For hardware implementation efficiency, reference [37] optimized the hidden layer calculation structure of neural networks, enabling self-tuning via a single field-programmable gate array (FPGA). In Reference [38], authors enhanced the input layer activation function by incorporating a nonlinear calculation factor, improving dynamic performance. The block diagram of the neural network PID controller with an enhanced nonlinear computation factor is presented in Figure 10.

In Reference [39], the RBFNN was integrated into the traditional PID algorithm for system identification, ensuring high precision, stability, and rapid response in multi-motor cooperative control. In reference [40], a feedforward neural network architecture was utilized to enhance the structure of the RBFNN, leading to the design of a self-tuning PID controller based on RBFNN. The resulting motor control system exhibited excellent tracking performance and strong anti-interference capabilities. The block diagram illustrating the structure of the RBFNN is presented in Figure 11.

In summary, scholars have employed optimization algorithms to refine the iterative process of neural network weights. This approach accelerates the self-tuning process of neural network control parameters. Additionally, scholars have enhanced the architecture of neural network PID controllers. These enhancements significantly improve the self-tuning performance of control parameters. The improved neural network PID controller methods and their corresponding advantages are presented in

Table 2. Enhanced methodology and associated advantages of the neural network PID controller.

Self-Tuning Method for Control Parameters	Enhanced Methods	Key Features
Neural network-based PID controller	Determine the initial values of weights	Enhanced efficiency
	Feedforward control approach	Streamline the calculation process
	Fuzzy logic algorithm	Acceleration of weight iteration
	Cuckoo search algorithm	Enhanced robustness
	Diagonally recursive optimization algorithm	Streamline the calculation process
	Adaptive compensator	Enhance overall accuracy
	Improved ANN	Enhanced efficiency
	Improved MPC	Enhanced dynamic performance
	Reinforcement learning with actor–critic methods	maneuverability and stability
	Optimize architecture of hidden layers	Optimization of hardware utilization
	Incorporate nonlinear computational factor	Enhanced dynamic performance

.

4. PID Controller Utilizing a Numerical Optimization Algorithm

The lightweight neural network is prone to converging on local optima, which compromises the performance of self-tuned PID control parameters. In contrast, deep neural networks involve substantial computational demands, extended iteration cycles, and stringent requirements for system hardware performance and storage capacity. Scholars have achieved commendable results by directly employing numerical optimization algorithms for the self-tuning of PID control parameters [41]. The architecture of a PID controller based on numerical optimization algorithms is illustrated in Figure 12.

On one hand, scholars have enhanced traditional numerical optimization methods such as GA and particle swarm optimization (PSO). For instance, Michael Neath et al. from the University of Auckland, New Zealand, utilized derivative-free optimization techniques to refine the traditional GA PID control parameter self-tuning system, leading to superior system performance [42]. Wang Hongzhi et al. from Changchun University of China introduced an improved GA-PID controller using quantum states as fundamental units. They effectively addressed the issue of local optima encountered by traditional GA-PID controllers due to improper selection, crossover, and mutation methods, thus ensuring stable motor speed control [43].

$$f(x,y)=-A\times \exp \left(-B\times \sqrt{{\displaystyle \frac{1}{d}}{\displaystyle \sum _{i=1}^d{x}^2}}\right)-\exp \left(\sqrt{{\displaystyle \frac{1}{d}}{\displaystyle \sum _{i=1}^d\cos (C\pi \times y^2)}}\right)+A$$

(17)

where A, B, and C represent the system gain constants, which are defined as per the literature. Additionally, A = 20, B = 0.2, C = 2, x and y denote the quantization characteristic parameters of the system state with x and y in (−10, 10).

Fan Xiaoshuai et al. from the National University of Defense Technology enhanced the traditional GA-PID controller via a nonlinear dynamic model, thereby improving its global search capability during parameter self-tuning [44]. The Vu Van Tan team at Hanoi University of Transport, Vietnam, optimized the PSO-PID controller within the home assistant (HASS) framework, which significantly improved the self-tuning ability of control parameters under nonlinear disturbances [45].

On the other hand, recognizing the limitations of traditional numerical optimization algorithms, various research teams have innovatively combined multiple traditional numerical optimization algorithms or proposed new optimization algorithms for control parameter self-tuning. In the literature [46], the combination of PSO and GA resulted in faster system step response. Reference [47] applied an integral anti-saturation strategy to the GA-PID control parameter self-tuning system, effectively resolving output saturation issues and suppressing harmonic components in current waveforms. Reference [48] integrated GA, PSO, Sparse Search Algorithm (SSA), and Differential Evolution (DE) to propose an improved monkey multiagent deep reinforcement learning (IMM-MADRL) optimization algorithm for PID control parameter self-tuning, achieving better overall performance. The block diagram of the IMM-MADRL optimization algorithm is presented in Figure 13.

Reference [49] introduced a cloud model-based quantum GA for PID control parameter self-tuning, overcoming the slow search speed of traditional GA through cloud crossover and mutation operators. Reference [50] evaluated the advantages and disadvantages of estimation of distribution algorithms, ant colony optimization, PSO, and model algorithms in PID control parameter self-tuning, proposing a metaheuristic algorithm with stronger anti-disturbance capabilities compared to traditional methods. In order to further reduce the complexity of the metaheuristic algorithm, the Technical University of Madrid [51] proposed a method for real-time adjustment of motor controller parameters by considering both performance and energy efficiency components in the objective function. This approach not only leverages the strong anti-disturbance capabilities of the metaheuristic algorithm, but also enhances the multi-objective optimization capability of the system. Reference [52] combined multiple input–output system control theory with real-valued GA to achieve simultaneous self-tuning of multiple PID controllers. Finally, reference [53] applied GA to optimize scaling factors and control rules of variable domain fuzzy inference, enabling online adaptive tuning of PID control parameters and significantly enhancing system control performance.

Reference [54] integrated the human learning mechanism with the Cauchy–Gaussian mutation strategy. An improved sparrow search algorithm for PID control parameter self-tuning was proposed, which not only strengthened the algorithm’s ability to avoid local optima, but also enabled it to effectively adapt to system changes and disturbances, thereby improving overall reliability. The mathematical representation of the improved sparrow search algorithm is presented in the following equation.

$$X_i^d(t+1)=X_{best}^d(t)+[X_i^d(t)-X_k^d(t)]\cdot [{\lambda }_{1}cauchy(0,{\sigma }^2)+{\lambda }_{2}Gauss(0,{\sigma }^2)]$$

(18)

In the formula, $X_i^d(t+1)$ denotes the position of the individual with the best fitness after mutation. $X_{best}^d\left(t\right)$ indicates the position of the current individual with the best fitness. $X_d^k$ represents an individual randomly selected from dimensions d and instance k. The parameters ${\lambda }_1$ and ${\lambda }_2$ are dynamically updated during the iterative process. The function Cauchy() signifies a random parameter drawn from a Cauchy distribution, while Gauss() signifies a random parameter drawn from a Gaussian distribution.

Reference [55] employs an adaptive weighted particle swarm optimization (AWPSO) algorithm to optimize the controller gain coefficient, thereby significantly enhancing the adaptability of the system. The motor control system achieves faster convergence, higher stability, and improved synchronization capabilities. The mathematical representation of the AWPSO algorithm is provided in the following equation.

$$\begin{array}{c}{V}^{k+1}=\omega V^k+c_1(P_{best}-P)rand(0,1)+c_2(g_{best}-P)rand(0,1)\\ P^{k+1}=P+V^{k+1}\end{array}$$

(19)

where V denotes the velocity of the particle, k represents the iteration index, P indicates the current position of the particle, rand(0, 1) refers to a random number uniformly distributed between 0 and 1, $c_1$ and $c_2$ are learning factor typically set to 2, and $\omega$ is the inertia weight ranging between 0.1 and 0.9.

Reference [56] employs a fuzzy logic PID controller, which demonstrates significantly enhanced control accuracy and efficiency compared to the conventional PID controller while also exhibiting superior system robustness and reliability. Reference [57] enhances the PID controller through PLIM closed-loop vector control, incorporating fuzzy logic to improve the responsiveness of torque control. In the literature [58], fuzzy logic is integrated with PSO, effectively preventing system overshoot and oscillation.

Due to the inherent randomness in the tuning results of the traditional fuzzy logic control parameter self-tuning method, the system is unable to rapidly acquire the optimal PID control parameters. In reference [59], the orthogonal optimization method is employed to predefine PID parameters, thereby enabling the system to swiftly and precisely select an optimal set of PID control parameters.

In summary, scholars have employed derivative-free optimization techniques, quantum properties, nonlinear dynamics algorithms, large-model inference acceleration algorithms, and other approaches to enhance traditional numerical optimization methods. These efforts have yielded significant improvements in the field of control parameter self-tuning. Additionally, scholars have innovatively improved traditional numerical optimization algorithms, effectively enhancing the performance of parameter self-tuning. The enhanced approach for the numerical optimization of the PID controller, along with its corresponding advantages, is presented in

Table 3. Numerical optimization algorithm for PID controller improvement and corresponding advantages.

Self-Tuning Method for Control Parameters	Enhanced Methods	Key Features
Numerical optimization algorithm	Derivative-free optimization method	Enhanced efficiency
	Enhanced algorithm for quantum property analysis	Avoid local optimal solution
	Nonlinear dynamic model	Avoid local optimal solution
	Inference algorithms for large-scale models	Minimize nonlinear disturbances
	Combination of PSO and GA	Enhanced step response speed
	Comprehensive anti-saturation strategy	Mitigate harmonic components
	Apply IMM-MADRL algorithm	Improved overall performance
	Quantum-inspired GA	Enhanced search efficiency
	Metaheuristic algorithm	Better resilience against disturbances
	Theory for multiple input–output systems	Multiple parameters self-tuning
	Fuzzy inference system	Enhanced system performance

.

In the actual industrial production process, single neurons and neural networks employ the gradient descent method to adjust control parameters in real-time. They are prone to becoming trapped in local optima, thus failing to obtain the optimal PID control parameters. It is possible to encounter multiple groups of local optimal solutions during the actual parameter tuning process of control systems.

PSO possesses the characteristics of global search capability, independence from gradients, relatively simple parameter adjustment, and suitability for parallel and efficient computation. It is particularly appropriate for modern control scenarios involving complex models and dynamic characteristics. As a numerical optimization algorithm, PSO finds extensive application in PID self-tuning engineering practices.

5. Simulation Experiment

To investigate the differences in the effectiveness of single-neuron, neural network, and numerical optimization algorithms for tuning control parameters, this study selects four scenarios, load torque mutation, large load moment of inertia, sudden velocity signal changes, and variable viscous friction coefficients, all under conditions where system electrical and mechanical parameters are unknown. The performances of the single-neuron PID control parameter self-tuning system, neural network PID control parameter self-tuning system, and PSO-based PID control parameter self-tuning system are compared. First, a parameter index reflecting system performance is defined, followed by simulations and comparisons of the aforementioned intelligent PID control parameter self-tuning methods conducted within the MATLAB 9.14 R2023a/Simulink simulation environment.

5.1. Data Collection Process

The resistance per phase is $R_S=2.2\mathsf{\Omega }$, the inductance along both the d-axis and q-axis is $L_d=L_q=3.95 \mathrm{m}\mathrm{H}$, the number of pole pairs is $p=4$, the flux linkage amplitude per phase is ${\phi }_f=0.1827 \mathrm{W}\mathrm{b}$, and the speed setpoint is 500 rpm. The motor has an initial moment of inertia of $J=0.00011 \mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$, a viscous friction coefficient of $B=0.0012 \mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$, and a rated torque of $T_N=1.0 \mathrm{N}·\mathrm{m}$. The electromagnetic and mechanical parameters of the motor utilized in the simulation experiment are presented in

Table 4. Simulation of motor electrical and mechanical parameters.

Motor Specifications	Numerical Data
Resistance of the motor stator ${\mathrm{R}}_{\mathrm{S}}/\mathsf{\Omega }$	2.2
Motor direct axis inductance ${\mathrm{L}}_{\mathrm{d}}/\mathrm{m}\mathrm{H}$	3.95
Motor quadrature axis inductance ${\mathrm{L}}_{\mathrm{d}}/\mathrm{m}\mathrm{H}$	3.95
Number of motor poles ${\mathrm{P}}_{\mathrm{n}}$	4
Flux of the permanent magnet motor ${\mathsf{\phi }}_{\mathrm{f}}/\mathrm{W}\mathrm{b}$	0.1827
Initial specified velocity ${\mathrm{v}}_{\mathrm{r}\mathrm{e}\mathrm{f}}/\mathrm{r}\mathrm{p}\mathrm{m}$	500
Moment of inertia $\mathrm{J}/$($\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$)	0.00011
Coefficient of viscous damping $\mathrm{B}/$($\mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$)	0.0012
Rated torque of the motor ${\mathrm{T}}_{\mathrm{N}}/$($\mathrm{N}·\mathrm{m}$)	1.3

.

The system following error is defined as follows:

$$e_k^i=x_d^i-x_k^i$$

(20)

In this context, i refers to the number of parameter self-tuning methods; $k$ represents the sampling time; $x_d^i$ signifies the desired speed; and $x_k^i$ indicates the actual speed.

Let $E$ represent the velocity tracking error metric, where the subscript $m$ denotes the root mean square (RMS) of the absolute error values. The $E_m$ values are as follows:

$$E_m=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=a}^{a+M}|e_k^i{|}^2}}$$

(21)

where $a \mathrm{a}\mathrm{n}\mathrm{d} M$ represent the initial sampling points and the length of the state interval in different states, respectively.

The single neuron and neural network are trained in an online manner. The input layer of the neural network consists of four neurons, which receive the actual motor speed, the reference speed, the speed error, and the integrated speed error with a forgetting factor as inputs. The hidden layer comprises seven neurons, and the activation function for these neurons is a symmetric Sigmoid function.

$$f(x)=\tanh \left(x\right)={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}$$

(22)

Backpropagation via gradient descent is employed to continuously update the network weights. The output layer contains three neurons, with an output range of 0 to 1. Since the three control parameters must remain non-negative, the activation function of the output layer neurons adopts a non-negative Sigmoid function.

$$g(x)={\displaystyle \frac{1}{2}}\left(1+\tanh \left(x\right)\right)={\displaystyle \frac{e^x}{e^x+e^{-x}}}$$

(23)

The performance index function is defined as the square of the control error.

$$E\left(k\right)={\displaystyle \frac{1}{2}}{\left(v_{ref}-v_{act}\left(k\right)\right)}^2$$

(24)

The tuned PID control parameters are obtained by multiplying the network output by the corresponding gain coefficients. The research team from Southeast University [60] have employed the Lyapunov method to investigate the stability of the automatic parameter tuning approach for the BPNN-PID controller. Their experimental findings confirm the efficacy of the proposed method and underscore the considerable improvement in the robustness and overall performance of the servo control system. The stability analysis of the BPNN-PID controller is not elaborated upon independently in this study.

In order to enhance the real-time performance of parameter tuning, as per Equation (19), the PSO algorithm was configured with 20 particles for each iteration in each generation. The inertia weight $\omega$ was set to 0.8, the learning factors were defined as $c_1$ = $c_2=2$, and the initial particle velocity was determined as 10% of the PID control parameter threshold.

The “To Workspace” module was utilized to gather speed data during the simulation experiment for subsequent analysis. The sampling period was established at 0.0001 s, and the total simulation time was set to 0.5 s.

5.2. Simulation Results

Under no-load conditions, the comparison of the simulated speed waveforms is presented in Figure 14, Figure 15 and Figure 16. The special symbol “*” in the figure legend denotes that the corresponding control signal is preset.

According to the simulation results, a comparison of the speed control error index was conducted among the single-neuron PID control parameter self-tuning method, the PID control parameter self-tuning method based on the PSO algorithm, and the neural network PID control parameter self-tuning method under no-load conditions. The comparisons are presented as follows. All results were differentiated based on the given speed. Results of the analysis are presented in Figure 17.

When a sudden load torque ($T_N=1.3 \mathrm{N}·\mathrm{m}$) is applied at 0.2 s, the comparison of the simulated speed waveforms is presented in Figure 18, Figure 19 and Figure 20. The special symbol “*” in the figure legend denotes that the corresponding control signal is preset.

After the application of a sudden load torque, the peak value of the absolute error between the actual speed and the reference speed is utilized as a metric to evaluate the impact of the abrupt load change on the control system’s performance. Results of the analysis are presented in Figure 21.

The load moment of inertia is configured to be 30 times the initial value, and the simulation speed waveform comparison is presented in Figure 22, Figure 23 and Figure 24. The special symbol “*” in the figure legend denotes that the corresponding control signal is preset.

According to the simulation results, the comparison of speed control error indices for the single-neuron PID control parameter self-tuning method, the PID control parameter self-tuning method based on the PSO algorithm, and the neural network PID control parameter self-tuning method under conditions of high inertia is presented below. Results of the analysis are presented in Figure 25.

When a speed command of $v_{ref}=700 \mathrm{r}\mathrm{p}\mathrm{m}$ is applied within the 0.2 s, the simulation speed waveform comparison is presented in Figure 26, Figure 27 and Figure 28. The special symbol “*” in the figure legend denotes that the corresponding control signal is preset.

According to the simulation results, a comparison of the speed control error index is conducted among three methods: single-neuron PID control parameter self-tuning, PID control parameter self-tuning based on the PSO algorithm, and neural network PID control parameter self-tuning under the condition of a sudden speed command change. The findings are presented as follows. Results of the analysis are presented in Figure 29.

The viscous friction coefficient has been adjusted to twice its original value, resulting in $B=0.0024 \mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$. Figure 30, Figure 31 and Figure 32 below present a comparison of the simulation speed waveforms under this new condition. The special symbol “*” in the figure legend denotes that the corresponding control signal is preset.

According to the simulation results, the data of each group were compared. Specifically, the speed control error index was evaluated for three methods: the single-neuron PID control parameter self-tuning method, the PID control parameter self-tuning method based on the PSO algorithm, and the neural network PID control parameter self-tuning method under the condition of a variable viscous friction coefficient, as shown below. All results were differentiated based on the given speed. Results of the analysis are presented in Figure 33.

In summary, the single-neuron system’s speed-following performance remains stable under sudden load torque and speed signals, but in no-load conditions, the speed-following error is relatively significant. Additionally, in systems with a large-load moment of inertia, there is a more pronounced overshoot in the speed-following system. In contrast, both the neural network PID controller and the PSO-PID controller demonstrate smaller speed-following errors under no-load conditions and exhibit high-intensity readings in systems with a large-load moment of inertia. However, after receiving load torque and speed signals, these controllers experience considerable fluctuations in speed. Adjustments to the system’s viscous friction coefficient have a minimal impact on the self-tuning performance of systems with three control parameters. The characteristics of three typical control parameter self-tuning methods are presented in

Table 5. Comparative analysis of three typical self-tuning methods for control parameters.

Self-Tuning Method for Control Parameters	Advantages	Limitations	Shared Characteristics
Neuron-PID	Minimal influence of abrupt load torque	Significant no-load following error	Minimal influence of viscous friction coefficient
	Minimal impact of sudden speed signal	Significant overshoot with large load
BPNN-PID	Minimal no-load following error	Significant influence of abrupt load torque
	Superior performance with large load	Significant impact of sudden speed signal
PSO-PID	Minimal no-load following error	Significant influence of abrupt load torque
	Superior performance with large load	Significant impact of sudden speed signal

.

5.3. Analysis of the Limitations of the Intelligent Self-Tuning Method

The performance disparities among three control parameter self-tuning algorithms should be attributed to variations in their computational complexities. References [61,62] have investigated the iterative computation process and computational complexity differences among single-neuron models, neural networks, and PSO algorithms. A single neuron consists solely of an input layer and an output layer, with its computation process involving a linear weighted sum, followed by direct output through an activation function. This approach does not require backpropagation calculations, only one matrix multiplication and activation function evaluation, resulting in microsecond-level computations per iteration. Consequently, its computational complexity is low, but limited to optimizing simple nonlinear systems or solving linearly separable problems. Neural networks involve forward propagation, which includes multi-layer matrix multiplications and activation functions, as well as backpropagation for gradient calculation and parameter updates. Backpropagation typically requires 2–3 times the computational effort of forward propagation. The overall computational cost scales proportionally with the number of layers and neurons, enabling the resolution of complex nonlinear problems, but at the expense of high computational demands that often necessitate high-performance computing resources such as GPU acceleration or parallel processing. Reinforcement-learning-based PID controllers exhibit notable limitations in this context. Reinforcement learning necessitates a substantial volume of interaction data to train the policy network, while industrial control systems demand high real-time performance, which precludes extensive trial-and-error processes. Furthermore, frequent adjustments to PID parameters can readily induce system oscillations. Additionally, deep reinforcement learning (DRL) requires the real-time execution of neural network inference, yet most PLCs or embedded controllers lack the computational power to meet these demands and achieve millisecond-level responses. The decision-making logic of neural networks is opaque, making it challenging for engineers to comprehend the rationale behind parameter adjustments, thereby failing to satisfy the interpretability requirements of industrial control systems. Finding the appropriate neural network learning rate and discount factor involves significant hyperparameter sensitivity, which can easily lead to differential effects and, consequently, result in the suboptimal self-tuning performance of control parameters.

The PSO algorithm searches for optimal solutions via particle swarm iterations, updating particle positions and velocities in each iteration while evaluating the fitness function. Its computational load scales proportionally with the number of particles and does not require gradient calculations. However, the optimization algorithm tends to converge to a local optimal solution, making it challenging to rapidly acquire the optimal PID control parameters.

Regarding guidance for user resource allocation, researchers, system designers, and practitioners are leveraging GPU acceleration, task scheduling, resource management, memory optimization, and parallel computing techniques to substantially reduce neural network training times. The pre-training of the neural network model enables the determination of a more suitable learning rate and network inertia coefficient, thereby mitigating the impact of differential effects. Single-neuron models and PSO algorithms exhibit short computation times per iteration and can operate efficiently on most industrial microcontrollers. In practical applications, engineers combine optimization algorithms with neural networks or single neurons to expand the range of control parameters and avoid becoming trapped in local optima.

6. Primary Research Challenges Discussion

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Nonlinearity and Coupling in PMSM Control Systems

Due to the presence of nonlinearities such as dead zones and saturation in the motor magnetic circuit within the control system’s inverter, as well as the coupling effects among the control axes of the multi-degree-of-freedom industrial servo control system, the early-stage control parameter self-tuning systems, such as single-neuron PID controllers and BP NN-PID controllers, have not adequately accounted for these factors. Consequently, this has led to reduced accuracy and compromised stability of the control system. Reference [63] utilized a deep reinforcement learning algorithm with continuous observation and action spaces to tune the proportional control parameters of nonlinear systems, enabling the rapid acquisition of optimal proportional control parameter values for PID controllers. Reference [64] proposed a cross-coupling control strategy that successfully resolved the issue of insufficient synchronization control accuracy in dual-motor servo systems under high load interference by incorporating active disturbance rejection control to enhance sliding mode control. Reference [65] developed an enhanced first-order active disturbance rejection controller based on an optimized nonlinear function, effectively addressing the challenges of poor control performance, synchronization, and stability in permanent magnet drive systems when using PID controllers for multi-motor control. Linear motors represent complex systems characterized by strong coupling, multi-variable dynamics, high-order nonlinearity, and time-varying behavior, which are susceptible to external disturbances and parameter mismatches. In reference [66], a sliding mode controller was designed using the exponential arrival law method, and sliding mode variable structure control was applied to the speed regulation system of a linear induction motor, achieving excellent control outcomes.

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Computational Complexity of PID Parameter Self-Tuning Algorithms

Because the utilization of advanced intelligent algorithms for self-tuning PID control parameters involves substantial computational requirements, it is often unsuitable for low-performance processors and fails to meet the demands of specific applications. Reference [67] introduced a novel Virtual Reference Feedback Tuning (VRFT) method combined with a Vector-Based Constant Amplitude Control (VCAC) algorithm to enhance the fuzzy logic optimization of PID controllers, significantly improving the efficiency of PID parameter self-tuning. Reference [68] proposed a fuzzy PID three-loop control strategy that notably decreased the computational load on DSPs during the control cycle compared to traditional self-adjusting PID controllers.

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Robustness and Adaptability

In industrial application scenarios, the load, temperature, and other environmental parameters of the PMSM control system frequently fluctuate, and motor parameters are prone to drift. This imposes stringent requirements on the adaptability of controller parameter self-tuning algorithms. Reference [69] integrates the parameter-solving process of PID control with the characteristics of multi-layer feedforward neural networks based on backpropagation algorithms to enhance the system’s robustness against external disturbances and uncertainties. Reference [70] proposes a control parameter tuning strategy based on PID improvement and optimal error performance, effectively improving the system’s capability to operate under various speed conditions. Reference [71] refines the second-order super-twisting sliding mode control algorithm using the recursive PID-NT (non-singular terminal sliding variable) estimation method, improving the accuracy and robustness of knee exoskeleton robots by incorporating advanced uncertainty mitigation techniques. Reference [72] combines generalized Type-2 fuzzy logic with memristor-based PID control to adaptively adjust controller parameters in real-time, significantly enhancing system robustness. Reference [73] merges the jellyfish optimization algorithm with traditional PID controllers, predicts motor running states via ANN, dynamically adjusts PID control parameters, adapts effectively to dynamic environmental conditions, and enhances system reliability.

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Multi-Objective Optimization Requirements

There may be multiple performance indices required by the PMSM control system, such as response speed, overshoot, and steady-state accuracy. It is necessary for multi-objective swarm intelligence optimization to balance conflicting performance indices and cross-paradigm controller designs to fully address the diverse and comprehensive requirements of industrial systems, including stability, energy efficiency, and real-time performance, ensuring optimal system functionality. Reference [74] enhanced the PID controller optimized by the GA by incorporating the Normal Distribution Crossover (NDX) operator, adaptive crossover probability, and variable neighborhood mutation mechanisms, thereby improving the performance of multiple objectives. To address the limitation of traditional PSO algorithms that are prone to falling into local optimal solutions, reference [75] introduced a multi-objective PSO algorithm for self-tuning PID control parameters, capable of satisfying multiple control objectives, such as controller constraints and output performance. In reference [76], the Multi-Objective State Transition Algorithm (MOSTA) was employed for self-tuning PID control parameters, demonstrating shorter computation times compared to the multi-objective PSO algorithm.

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Experimental validation and hardware performance evaluation

Currently, the self-tuning methods for PID controller parameters proposed by many scholars remain primarily in the simulation verification stage and lack sufficient physical experimental validation. The practical application effectiveness of these proposed self-tuning methods requires further detailed analysis. In reference [77], the Variable Neighborhood Search (VNS) algorithm was employed to achieve self-tuning of the control parameters for a PID controller. A system performance hardware test based on an STM32 development board confirmed the efficacy of the control parameter self-tuning algorithm. In reference [78], the self-tuning of PID controller parameters was conducted using system identification and adaptive deep learning algorithms, with the performance of the self-tuning algorithm being evaluated within an actual industrial system. Reference [79] enhanced the fuzzy optimization PID controller parameter self-tuning method based on the resonant frequency of the control vector, validating the proposed algorithm’s effectiveness on a transducer experimental platform. An improved multi-stage hysteresis and fuzzy controller-based predictive direct torque control (MPDTC) application-specific integrated circuit (ASIC) was proposed in reference [80]. The hardware architecture was implemented using Verilog Hardware Description Language (HDL), and the ASIC was fabricated using TSMC 0.18 μm 1P6M CMOS technology. Compared to traditional DTC systems designed using FPGA development boards, hardware testing demonstrates that this scheme exhibits lower energy consumption as well as superior robustness and convenience.

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Insufficient integration with traditional control methodologies

The hybrid architecture, which integrates modern intelligent methods with classical frequency domain adjustment techniques, represents a design concept that merges data-driven models with prior signal processing knowledge. However, over the past three years, research on the application of hybrid architectures in the motor domain has predominantly focused on motor fault diagnosis. Correspondingly, there is a noticeable lack of studies exploring the use of hybrid architectures in motor control, particularly in the realm of PID control parameter tuning. In reference [81], a novel hybrid adaptive fusion deep learning model was introduced by combining synchronous compressed short-time Fourier transform with Kolmogorov–Arnold network, significantly enhancing the accuracy of motor diagnosis under high-noise conditions. Reference [82] proposed an innovative fault diagnosis method, equipping a multi-input convolutional neural network (MICNN) with a feature extraction model and further refining this model through Bayesian deep learning to quantify the uncertainty of motor fault diagnosis results and improve the interpretability of the model. Reference [83] introduced an Adaptive Local Binary Pattern (ALBP) method based on traditional Local Binary Pattern (LBP) technology and convolutional neural networks (CNNs), further elevating the precision of motor fault diagnosis.

7. Conclusions

There is a scarcity of retrospective studies in the field of motor controller parameter self-tuning. While some scholars have reviewed artificial intelligence-based self-tuning algorithms for motor control parameters, there remains a gap in the application of numerical optimization algorithms for control parameter self-tuning. Additionally, no simulation comparison has been conducted to analyze the performance differences between numerical optimization algorithms and artificial intelligence algorithms in the context of control parameter self-tuning. This study addresses these research gaps by providing a comprehensive simulation analysis of the effects of single-neuron, neural network, and PSO algorithms on control parameter tuning under four conditions: load torque mutation, high-load moment of inertia, sudden speed signal changes, and variable viscous friction coefficients. The findings provide valuable insights into the practical applications of control parameter self-tuning algorithms. Furthermore, this paper examines the research challenges and key achievements in intelligent self-tuning technology for PID control parameters, summarizing the future development trends and challenges as follows:

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Enhanced control accuracy: By employing more optimized control methods and advanced control parameter self-tuning algorithms, the tracking precision, response speed, and computational efficiency of the control system can be further improved.

(2)

Greater algorithm and framework integration: A hybrid control parameter self-tuning method that combines numerical optimization algorithms, neural networks, and single-neuron approaches is proposed to leverage the strengths of different algorithms across various application scenarios. The neural network weights are optimized using the application of a numerical optimization algorithm. A hybrid control parameter self-tuning framework that integrates intelligent algorithms with traditional frequency domain analysis and state observers can effectively leverage the strengths of various control parameter self-tuning methods, thereby achieving superior tuning performance.

(3)

Improved adaptability: Advanced control theory and optimization algorithms are utilized to balance multiple performance indices, ensuring that the servo system operates at maximum efficiency while achieving high performance.

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Increased specificity: A targeted controller parameter self-tuning method is developed based on the specific requirements and characteristics of actual system operating conditions.