Improved Grey Wolf Optimizer and Backpropagation Neural Network for Fractional-Order Control of PMSM

Abstract

Permanent magnet synchronous motors (PMSM), with their high efficiency and power density, are widely used in industrial applications. For PMSM speed servo systems, fractional-order proportional-integral (FOPI) controllers demonstrate superior robustness and speed control performance compared to conventional PI controllers. However, FOPI controllers involve more parameters with insufficient tuning experience, making their parameter design more challenging. To address the aforementioned problems, this paper proposes a novel FOPI control strategy for PMSM that integrates an improved grey wolf optimizer (IGWO) with backpropagation neural networks (BPNNs). The conventional grey wolf optimizer (GWO) is enhanced in this study, and the test results demonstrate that the proposed IGWO exhibits improved convergence and robustness. BPNNs combined with IGWO are employed to fit the relationship between controller parameters and control performance indicators. IGWO employs the fitted values from BPNNs as evaluation criteria to perform online tuning of the FOPI controller parameters. The simulation results show that under the IGWO-BPNN-FOPI control strategy, the overshoot of the PMSM speed step response is only 1.958%, the settling time is reduced by 80%, and the load disturbance rejection performance of the PMSM is significantly improved.

1. Introduction

Permanent magnet synchronous motors (PMSM) are widely used in new energy vehicles and engineering machinery due to their advantages of high speed, high efficiency, and low noise [1,2]. To achieve robust control of PMSM, various control schemes have been proposed in the industry [3,4,5]. Among these, the application of fractional-order proportional–integral–derivative (FOPID) control in PMSM speed servo systems has become a prominent research direction, driven by advances in fractional calculus and fractional-order systems [6,7]. Podlubny [8] proposed a FOPID control theory by introducing the integral order λ and derivative order μ to the conventional PID controller, which provides a new perspective for PID control methodologies. Through continuous development and practice, FOPID controllers have demonstrated better robustness and superior control performance compared to traditional PID controllers [9,10]. However, FOPID controllers involve more parameters, making tuning more challenging, and the influence mechanisms of the λ and μ on the control performance are difficult to determine. Suitable and innovative methods are required to address the design challenges of FOPID controllers. Current design methods for FOPID controllers can be classified into frequency- and time-domain-based methods.

Frequency-domain-based methods design controller parameters according to the characteristics of the control system to meet predefined frequency-domain specifications [11,12]. For example, Chevalier et al. [13] proposed a three-parameter FOPID controller, which determines the three parameters by establishing relationships between the integral gain and derivative gain, as well as between the integral order and derivative order. Chen et al. [14] developed a frequency-domain-based parameter design method for FOPID controllers specifically for PMSM systems. The controller design method proposed in their study offers broader applicability and satisfies the requirements for speed response and robustness in PMSM systems.

Time-domain-based methods iteratively optimize the controller parameters according to the time-domain specifications of the control system [15]. With the development of modern metaheuristic optimization methods, an increasing number of optimization algorithms have been applied to the parameter tuning of FOPID controllers. Jeetendra et al. [16] applied the grey wolf optimizer (GWO) to a DC motor FOPID controller, demonstrating significantly better control performance for motor speed compared to traditional PID control methods. Hekimoglu et al. [17] proposed a chaotic atom search optimization (ChASO) algorithm incorporating chaotic sequences and applied it to a DC motor FOPID speed controller. By combining Bode diagrams for frequency-domain response analysis, the study demonstrated the effectiveness of ChASO in FOPID controller parameter tuning. Xuanru et al. [18] applied an improved sparrow search algorithm to the parameter optimization of FOPID controllers, showing that this enhanced optimization method is more suitable for fractional-order PID parameter tuning. Sime et al. [19] proposed a novel genetic algorithm-optimized adaptive fuzzy FOPID controller, which improves the speed control performance of electric vehicle PMSM drive systems.

Among the numerous metaheuristic optimization methods, the GWO [20] is known for its simple structure and fast convergence speed, and has been widely applied across various fields [21,22,23,24]. However, GWO still suffers from limitations such as dependence on the initial population [25] and an imbalance between global exploration and local exploitation capabilities [26]. To enhance the suitability of GWO for fractional-order proportional-integral (FOPI) controller parameter optimization, this study introduces improvements in three key aspects of the algorithm.

In previous parameter optimization processes for FOPI controllers, evaluating the quality of parameter sets required inputting them into the controlled system to obtain control performance indicators. To rapidly acquire these performance indicators, such methods often simplify the controlled system into a directly representable transfer function form [15]. However, considering factors such as temperature rise effects, load disturbances, and DC bus voltage fluctuations, the PMSM represents a relatively complex system. Deriving an accurate transfer function for such a system is challenging, and a simplified transfer function may not adequately represent an actual controlled system. To bypass the computational process of deriving the transfer function, this study employs backpropagation neural networks (BPNNs) to fit the mapping relationship between the controller parameters and control performance indicators in the PMSM speed servo system. This approach enables rapid evaluation of the quality of the current controller parameters.

The major contributions of this paper are as follows: (1) This paper improves the traditional GWO. We propose a new nonlinear convergence factor and a stochastic position update formula, which improve the convergence and stability of GWO. We apply Chebyshev chaotic mapping for population initialization in GWO to enhance population diversity. (2) Based on IGWO, this paper designs a training method for BPNNs. The BPNNs successfully fit the relationship between the controller parameters and control performance indicators in the PMSM speed servo system, thereby achieving sufficient fitting accuracy. (3) Combining IGWO and BPNN, this paper proposes an FOPI control strategy. Simulation results verify that this control strategy exhibits superior speed regulation performance and enhanced load disturbance rejection capability.

2. PMSM Speed Servo System Model

This paper focuses solely on the speed control of a PMSM without considering its specific operational states. Given the complexity of actual PMSM operating conditions, the following assumptions are made to facilitate the analysis of its mathematical model [27]:

(1)

The magnetic circuit is unsaturated, meaning the motor’s inductance is not affected by current changes. Eddy current and hysteresis losses are ignored.

(2)

The effects of the slotting, commutation processes, and armature reactions are neglected.

(3)

The three-phase windings are fully symmetric, and the magnetic field of the permanent magnet follows a sinusoidal distribution around the air gap.

(4)

The armature windings are evenly and continuously distributed on the inner surface of the stator, the three-phase windings are star-connected, and the sum of the three-phase currents is zero.

(5)

Neglect measurement noise.

(6)

The actuator saturation effect is considered, meaning the output current is limited within the physically achievable maximum range.

(7)

Computational delays are neglected, and fixed-step sampling is employed.

Under the aforementioned assumptions, and based on the SVPWM vector control principle, this study establishes a permanent magnet synchronous motor speed servo system simulation model in MATLAB/Simulink, as shown in Figure 1. The version of MATLAB we used is R2019b. Both the inverter module and the PMSM module in the aforementioned simulation system are implemented using models from the toolbox in Simulink. The inverter module is a typical two-level three-phase voltage source inverter circuit, which can directly receive signals from the SVPWM module and supply voltage to the motor’s stator windings. Different preset PMSM models in Simulink can be selected according to specific requirements. The motor selected in this paper has a rated torque of 26.13 Nm, a rated speed of 3000 r/min, and a rated DC link voltage of 560 V. The PMSM module can output torque to external systems while simultaneously providing signals such as real-time stator phase currents and rotor position.

3. Improved Grey Wolf Optimizer

3.1. Introduction to the Traditional Grey Wolf Optimizer

GWO is a bio-inspired algorithm proposed by Mirjalili et al. [20] in 2014. The GWO algorithm simulates the leadership hierarchy and hunting mechanisms of grey wolves in nature. As shown in Figure 2, the grey wolf population is divided into a strict four-level social hierarchy: α wolf, β wolf, δ wolf, and ω wolves. The α wolf, serving as the leader responsible for decision-making, corresponds to the optimal solution; the β wolf assists the α and represents the suboptimal solution; the δ wolf acts as scouts and sentinels, corresponding to the third-best solution; while the ω wolf comprises ordinary members of the pack, representing other candidate solutions. The first three wolves function as leaders that guide the rest of the pack in locating and attacking prey. The algorithm treats the prey’s position as the ultimate objective, simulating the three main phases of grey wolf behaviour: searching for prey, encircling prey, and attacking prey.

After determining the number of grey wolves N in the wolf pack and the dimension D of the problem to be solved, it is necessary to initialize the position of each grey wolf. The position vector of the grey wolf individual n at the t-th iteration is given by Equation (1):

$$X_n^t=\left[\begin{array}{cccccc}{x}_1& x_2& \dots & x_d& \dots & x_D\end{array}\right]$$

(1)

where t is the current iteration number. Meanwhile, n is the grey wolf identifier and $x_d$ is the value of the grey wolf’s position vector in the d-th dimension.

The fitness of each grey wolf is calculated based on the specific problem to be solved. Based on the fitness values, the top three best-performing grey wolves are selected as the α wolf, β wolf, and δ wolf, with their position vectors denoted as X_α, X_β, and X_δ, respectively. The remaining wolves are classified as ω wolves.

The hunting (solution search) process is guided by the top three wolves, while the remaining grey wolves conduct their search based on the positions of these three leaders. During the prey search (position update), each grey wolf individual needs to calculate its distance to the current α, β, and δ wolves according to Equation (2).

$$\left\{\begin{array}{l}{D}_{\alpha }=\left|C_{\mathrm{a}}·X_{\alpha }-X_n^t\right|\\ D_{\beta }=\left|C_{\mathrm{b}}·X_{\beta }-X_n^t\right|\\ D_{\delta }=\left|C_{\mathrm{c}}·X_{\delta }-X_n^t\right|\end{array}\right.$$

(2)

where D_α, D_β, and D_δ represent the distances between the individual grey wolf and the current α, β, and δ wolves, respectively. X_n^t denotes the position of the grey wolf individual at the t-th iteration. C_a = 2$r_1$, C_b = 2$r_2$, and C_c = 2$r_3$, where $r_1$, $r_2$, and $r_3$ are random numbers uniformly distributed in the range [0, 1]. These three coefficients are used to simulate the randomness of grey wolves’ movement around the prey.

To simulate the grey wolves’ estimation of the prey’s position and their corresponding movement during the hunting process, the positions of the ω wolves are updated based on the distances calculated from Equation (2) as follows:

$$\left\{\begin{array}{l}{X}_a=X_{\alpha }-A_a·D_{\alpha }\\ X_b=X_{\beta }-A_b·D_{\beta }\\ X_c=X_{\delta }-A_c·D_{\delta }\\ X_n^{t+1}={\displaystyle \frac{X_a+X_b+X_c}{3}}\end{array}\right.$$

(3)

where X_a, X_b, and X_c are the adjusted positions of one ω wolf influenced by the α, β, and δ wolves, respectively. The new position X_n^t+1 of the grey wolf is obtained by taking the average of these three values. A_a, A_b, and A_c are dynamic coefficients that determine the intensity and direction of the grey wolf’s movement toward the prey. The values of these three coefficients affect how closely the grey wolves encircle the prey, thereby influencing the exploration and exploitation characteristics of the search behaviour. A_a, A_b, and A_c are calculated as follows:

$$\left\{\begin{array}{l}{A}_a=a·(2r_4-1)\\ A_b=a·(2r_5-1)\\ A_c=a·(2r_6-1)\end{array}\right.$$

(4)

$$a=2\left(1-{\displaystyle \frac{t}{T}}\right)$$

(5)

In Equation (4), $r_4$, $r_5$, and $r_6$ are random numbers within [0, 1], and a is the convergence factor. As shown in Equation (5), a decreases from 2 to 0 during the iteration process. This reduction simulates the behaviour of grey wolves gradually approaching their prey during the hunting process.

3.2. Improvements to the Grey Wolf Optimizer

3.2.1. Initialization Based on Chebyshev Chaotic Mapping

Chebyshev chaotic mapping [28] exhibits excellent pseudorandom characteristics and is highly sensitive to initial parameters. Minute differences are amplified exponentially, ensuring significant diversity among population individuals. Compared to random uniform distribution, chaotic sequences can explore the search space more uniformly, preventing the initial population from clustering in local regions and reducing the risk of premature convergence. Therefore, this paper employs the Chebyshev chaotic mapping to initialize the wolf population. The formula for the Chebyshev chaotic mapping is as follows:

$$\begin{array}{cc}{x}_{n+1}=\cos \left(K·\arccos \left(x_n\right)\right),& x_n\in \left[-1,1\right]\end{array}$$

(6)

where K denotes the order, and here K is set to 4. At this order, regardless of how similar the initial values are, the resulting iterative sequences remain mutually uncorrelated. This ensures that the sequences exhibit chaotic and ergodic properties within this parameter range.

Based on the Chebyshev mapping, the wolf population initialization steps are as follows:

(1)

Set the number of grey wolves to N, and randomly generate the initial vector of the first grey wolf individual Y₁¹ = [$y_{\mathrm{1,1}}$… $y_{1,d}$… $y_{1,D}$], where each element $y_{1,d}$ is a random number within the range [−1, 1].

(2)

Use Equation (6) to perform N − 1 iterations on each dimension of Y₁¹, generating the initial vectors for the remaining N − 1 grey wolf individuals.

(3)

Determine the upper bound $u_d$ and lower bound $l_d$ of the d-th dimension in the search space, and then use the following equation to generate the position vectors of the N grey wolf individuals:

$$x_{n,d}=l_d+Sigmoid\left(5y_{n,d}\right)·\left(u_d-l_d\right)$$

(7)

where $y_{n,d}$ is the initial value of the n-th wolf in the d-th dimension. After multiplying by 5, its range expands to [−5, 5], which allows it to more closely approach the upper and lower bounds of the Sigmoid function. $x_{n,d}$ is the coordinate value of the n-th wolf in the d-th dimension of the search space. The mathematical expression of the Sigmoid function is as follows:

$$Sigmoid(x)={\displaystyle \frac{1}{1+e^{-x}}}$$

(8)

The purpose of using the Sigmoid function is to make the initial positions of the grey wolves distribute more intensively near the boundaries of the search space. This ensures a more thorough and comprehensive search of the entire space by the wolf pack, preventing the oversight of optimal values that may be located near the boundaries.

3.2.2. Nonlinear Convergence Factor

In Equation (3), the value of A influences the exploitation capability of GWO, while changes in the convergence factor a directly affect the value of A. In the traditional GWO, the convergence factor decreases linearly from 2 to 0. Sasmita et al. [21] demonstrated that a linearly decreasing convergence factor fails to effectively distinguish between global search and local search. In GWO, the search space is extensive during the initial stages, necessitating rapid exploration; however, in the later stages, when closer to the prey, an excessively large step size may lead to overshooting the search space and prematurely converging to local optima. Therefore, a fine-grained search is required in the final phase.

Based on the aforementioned considerations, this paper proposes a nonlinear convergence factor, whose mathematical expression is given as follows:

$$a=4\left[1-{\left(1{+\mathrm{e}}^{{\displaystyle \frac{\mathrm{k}}{\ln \left(\frac{t-1}{T}\right)}}}\right)}^{-1}\right]$$

(9)

where k is the convergence coefficient. Figure 3 shows the variation in the convergence factor a during the GWO search process when k takes different values. A larger value of k results in a slower decrease in a in later stages. In the early stages of iteration, it decreases rapidly, enabling quick discovery of potential extreme values during global search. In the later stages, the rate of decrease slows down, and the reduced search step size helps prevent missing the optimal solution. To balance the rapid search in the early stage and the refined optimization in the later stage, k is set to 2.

3.2.3. Stochastic Position Update Formula

In GWO, the excessively strong guiding influence of the α, β, and δ wolves can lead to premature aggregation of the population. This occurs because the position updates of the wolf pack rely too heavily on the leading wolves, resulting in a loss of population diversity. Under these circumstances, the overly rapid convergence of GWO may cause the algorithm to become trapped in local optima or miss the global optimal solution. To address these issues, the position update formula in Equation (3) is improved as follows:

$$\left\{\begin{array}{l}{X}_a=X_{\alpha }-A_a·D_{\alpha }\\ X_b=X_{\beta }-A_b·D_{\beta }\\ X_c=X_{\delta }-A_c·D_{\delta }\\ X_n^{t+1}={\displaystyle \frac{r_7·X_a+r_8·X_b+r_9·X_c}{3}}\end{array}\right.$$

(10)

where $r_7$, $r_8$, and $r_9$ are random numbers uniformly distributed in the range [0, 1]. By incorporating these three random numbers into the original formula, the dependence of individual grey wolves on the leading wolves can be reduced.

3.3. Performance Comparison of Optimization Algorithms

To verify the effectiveness of the improvements made to the grey wolf optimization algorithm, we select six benchmark functions [29] as objective functions to test three algorithms: IGWO, GWO [16], and ChASO [17]. The testing process involves using these three algorithms to search for the global minimum values of these functions. The definitions of these benchmark functions are summarized in

Table 1. Benchmark functions.

Fun	Mathematical Expression
F1	$f\left(x\right)={\displaystyle \sum _{i=1}^d{x}_i^2}$
F2	$f\left(x\right)={\displaystyle \sum _{i=1}^{d-1}[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2]}$
F3	$f\left(x\right)={\displaystyle \sum _{i=1}^d{\left(x_i+0.5\right)}^2}$
F4	$f\left(x\right)={\displaystyle \sum _{i=1}^d\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]}$
F5	$f\left(x\right)=-20\exp \left(-0.2\sqrt{{\displaystyle \frac{1}{d}}\pm {\displaystyle \sum _{i=1}^d{x}_i^2}}\right)-\exp \left({\displaystyle \frac{1}{d}}{\displaystyle \sum _{i=1}^d\left(\cos 2\pi x_i\right)}\right)+20+e$
F6	$f\left(x\right)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^d{x}_i^2}-{\displaystyle \prod _{i=1}^d\cos \left(\frac{x_i}{\sqrt{i}}\right)+1}$

. The benchmark functions all have a dimensionality of 30, and their global minimum values are all 0.

The population size for each algorithm is set to 20, and the maximum number of iterations is set to 50. To mitigate errors caused by random population initialization, each algorithm is independently run 30 times, and the average value (MEAN) and standard deviation (STD) are taken as comparison parameters. The convergence curves of each algorithm under different functions are shown in Figure 4. It can be observed from the figure that IGWO exhibits significantly faster convergence speed, demonstrating clear advantages when the number of iterations is limited. The test results are presented in

Table 2. Algorithm test results.

Fun	Metric	IGWO	GWO	ChASO
F1	MEAN	0	3.989	677.335
	STD	0	2.001	188.368
F2	MEAN	28.960	2.067 × 103	2.952 × 105
	STD	0.017	1.418 × 103	1.167 × 105
F3	MEAN	6.285	7.890	645.718
	STD	0.417	2.634	176.973
F4	MEAN	0	155.538	926.846
	STD	0	47.936	177.958
F5	MEAN	8.882 × 10−16	3.373	13.227
	STD	0	0.240	1.007
F6	MEAN	0	0.056	1.075
	STD	0	0.052	0.021

. The MEANs of IGWO are noticeably smaller than those of other optimization algorithms, indicating that IGWO can achieve higher precision results under the same number of iterations. Moreover, the STDs of IGWO are lower than those of other algorithms, which confirms that IGWO possesses superior stability.

4. FOPI Control Strategy

4.1. Implementation of Discrete FOPI Control

The FOPID controller incorporates one additional integration order λ and one additional differentiation order μ compared to the conventional PID controller. Its transfer function can be represented as:

$$C\left(s\right)=K_p+{\displaystyle \frac{K_i}{s^{\lambda }}}+K_d{s}^{\mu }$$

(11)

where K_p is the proportional coefficient, K_i is the integral coefficient, and K_d is the derivative coefficient. When μ = 0, it becomes an FOPI controller.

The fractional-order PID controller is implemented based on fractional calculus. To better describe fractional differentiation and integration, a unified fractional calculus operator [7] is introduced here:

$${}_{t_0}D_t^{\alpha }f\left(t\right)=\left\{\begin{array}{l}\begin{array}{cc}{\displaystyle \frac{d^{\alpha }}{d{t}^{\alpha }}}f\left(t\right),& \alpha >0\end{array}\\ \begin{array}{cc}f\left(t\right),& \alpha =0\end{array}\\ \begin{array}{cc}{\displaystyle {\int }_{t_0}^{t}f\left(\tau \right)d{\tau }^{-\alpha },}& \alpha <0\end{array}\end{array}\right.$$

(12)

where α is restricted to real numbers, t is the independent variable, and t₀ is the lower boundary of the independent variable. When α > 0, it represents the α-order derivative of the function with respect to the independent variable t; when α = 0, it denotes the original signal; when α < 0, it represents the−α-order integral. If α ≥ 0 and t₀ = 0, t₀ can be omitted. If the independent variable is t and there are no other variables, t can also be omitted.

The Grünwald–Letnikov definition of the α-order derivative of the function f(t) is given by:

$${}_{t_0}{}^{GL}D_t^{\alpha }f\left(t\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{1}{h^{\alpha }}}{\displaystyle \sum _{j=0}^{\left[\left(t-t_0\right)/h\right]}{\left(-1\right)}^j\left(\begin{array}{c}\alpha \\ j\end{array}\right)f\left(t-jh\right)}$$

(13)

where [(t − t₀)/h] denotes taking the nearest integer. When the step size h is sufficiently small, the Grünwald–Letnikov definition can be expressed as follows:

$${}_{t_0}{}^{GL}D_t^{\alpha }f\left(t\right)\approx {\displaystyle \frac{1}{h^{\alpha }}}{\displaystyle \sum _{j=0}^{\left[\left(t-t_0\right)/h\right]}{w}_{j}f\left(t-jh\right)}$$

(14)

where w_j is the coefficient of the binomial term (1 − z)^α, which can be obtained recursively as follows:

$$\begin{array}{cc}{w}_0=1,& w_j=\left(1-{\displaystyle \frac{\alpha +1}{j}}\right)w_{j-1}\end{array}$$

(15)

Microcontrollers are incapable of processing signals continuously in real time during actual operation; instead, they acquire and output signals at fixed time intervals. To more closely approximate real-world conditions, the PMSM model in Section 2 is simulated with a fixed time step discretization. To maintain consistency with the aforementioned simulation model, the FOPI controller must also be discretized. Based on Equations (14) and (15), the discretization of the FOPI controller is implemented as follows:

(1)

Let the sampling time be T_sa, then at the k-th sampling point, the controller’s operating time is t = kT_sa. Assuming the error at time t is e(k), and the fractional-order integral is D^−λe(k), the output of the FOPI controller at time t can be expressed as follows:

$$u\left(k\right)=K_{p}e\left(k\right)+K_i{D}^{-\lambda }e\left(k\right)$$

(16)

(2)

By replacing α with −λ in Equation (15), the binomial coefficient sequence w(k) can be calculated. Then, the computation of D^−λe(k) in Equation (16) is given by:

$$D^{-\lambda }e\left(k\right)=T_{s\mathrm{a}}^{-\lambda }{\displaystyle \sum _{i=1}^{k}w\left(i\right)e\left(k-i+1\right)}$$

(17)

4.2. BPNN Based on IGWO

The ultimate objective of this study is to utilize IGWO for iterative optimization of the FOPI controller parameters. Conducting a simulation in the model for each set of controller parameters at every iteration would be highly time-consuming. To reduce temporal costs and bypass the computational process of deriving the PMSM system’s transfer function, this paper directly employs BPNNs to fit the mapping relationship between the controller parameters and performance indicators in the PMSM speed servo system. These performance metrics are then used to evaluate the quality of the current controller parameters. The control performance indicators selected in this paper are response time, overshoot, settling time, and the integral of time multiplied by absolute error (ITAE). The calculation formula for ITAE is shown in Equation (18).

$$ITAE={\displaystyle {\int }_0^{+\infty }t\left|e(t)\right|dt}$$

(18)

Traditional BPNN employs the gradient descent method for weight updates, which suffers from slow convergence speed and a tendency to fall into local optima [30]. Therefore, we utilize the proposed IGWO to update the weights and biases of the BPNN. The first step involves mapping the weight matrices and bias matrices between each layer of the BPNN to the position vectors of the wolf pack:

$$W_n^t=\left[\begin{array}{cccccccc}{w_{1,1}}^1& {w_{1,2}}^1& \dots & {w_{i,j}}^c& {{\theta }_1}^1& {{\theta }_2}^1& \dots & {{\theta }_h}^c\end{array}\right]$$

(19)

where w_i,j^c is the connection weight from the i-th node of the c-th layer to the j-th node of the c+1-th layer, while θ_h^c is the bias (threshold) of the h-th node in the c+1-th layer.

This study finds that a single complex BPNN outputting multiple performance indicators demonstrates inferior fitting effectiveness compared to multiple simpler BPNNs, each dedicated to outputting a single performance indicator. Consequently, four distinct BPNNs are designed targeting the following performance indicators: response time (T_r), overshoot (M_p), settling time (T_s), and integral of time multiplied by absolute error (ITAE). The specific network structures are presented in

Table 3. Structure of the backpropagation neural networks.

Output Value	Number of Inputs	Number of Neurons in Hidden Layer 1	Number of Neurons in Hidden Layer 2	Number of Outputs
Tr	3	4	4	1
Mp	3	6	6	1
Ts	3	5	5	1
ITAE	3	9	9	1

. In response to the highly nonlinear characteristics of the fractional-order PMSM control system and to capture its hierarchical dynamic features, we adopted a dual-hidden-layer architecture. The input values for these four networks are the parameters of the FOPI controller: K_p, K_i, and λ. Therefore, the number of neurons in the Input Layer for each BPNN is 3. Since each BPNN is designed to fit a single performance metric, the number of neurons in the Output Layer for each BPNN is 1. Based on common practice, we initially set the number of Hidden Layer neurons to 1.5 times the number of input features, and then gradually increased the neurons until the performance no longer improved significantly. The activation functions for both Hidden Layer 1 and Hidden Layer 2 are the Tanh function, while no activation functions are applied to the Input Layer and Output Layer. The calculation formula for the Tanh function is given in Equation (20).

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

(20)

The data for the neural network is derived from the simulation model described in Section 2, comprising a total of 210 data points. The data in the dataset is randomly distributed. The first 80% of the data is used as training data, while the remaining 20% serves as testing data. The training process of the BPNN based on IGWO (IGWO-BPNN) is shown in Figure 5. The average fitting errors of the BPNNs for each performance indicator on the test data are presented in

Table 4. The fitting errors of the BPNNs for performance indicators.

Performance Indicators	Average Fitting Error
Tr	19%
Mp	3%
Ts	21%
ITAE	25%

. The formula for calculating the average fitting errors (AFE) is shown in Equation (21):

$$AFE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{P_i-D_i}{D_i}\right|}$$

(21)

where D_i is the true value of the i-th test sample, P_i is the prediction of the BPNN for the i-th test sample, and n is the number of test samples.

The fitting results are shown in Figure 6. The reason for the larger average fitting error may be that some data points have extremely small values, leading to the scenario of dividing a large number by a small number when calculated using Equation (21). Consequently, a few individual data points with large errors have affected the overall average. This may affect the subsequent optimization search by the IGWO to some extent. However, as long as the BPNN’s predictions can effectively distinguish between larger and smaller values, it will not impact the final assessment of the current controller parameters’ quality. The fitting accuracy meets the requirements for evaluating the performance of the current controller parameters.

4.3. FOPI Control Strategy Based on IGWO and BPNN

The principle of the FOPI control strategy based on IGWO and BPNN (IGWO-BPNN-FOPI) is illustrated in Figure 7. The IGWO algorithm is used to perform online tuning of the FOPI controller parameters: K_p, K_i, and λ. For each parameter set, the control performance indicators are obtained through the pre-trained BPNN from Section 4.2, and then the fitness of each wolf is calculated. The fitness function designed in this study is as follows:

$$Fitness\left(n\right)={\left[K_1{\left({\displaystyle \frac{T_r}{T_{r\_mu}}}\right)}^2+K_2{\left({\displaystyle \frac{M_p}{M_{p\_mu}}}\right)}^2+K_3{\left({\displaystyle \frac{T_s}{T_{s\_mu}}}\right)}^2+K_4{\left({\displaystyle \frac{ITAE}{ITA{E}_{mu}}}\right)}^2+{\left({\displaystyle \frac{error}{setting\_error}}\right)}^2\right]}^{-1}$$

(22)

where T_{r_mu}, M_{p_mu}, T_{s_mu}, and ITAE_mu represent the mean values of the corresponding performance indicators from the BPNN training data in Section 4.2, respectively. K₁, K₂, K₃, and K₄ are the weights assigned to the response time, overshoot, settling time, and ITAE in the fitness function, respectively. Different weight configurations will lead to different control effects. To suppress overshoot, we set K₁ = 1, K₂ = 6.5, K₃ = 1, and K₄ = 1. error is the difference between the reference speed and the actual speed; setting_error is 2% of the reference speed.

The wolf with the highest fitness is selected, and its position vector is used as the current controller parameters. The three wolves with the lowest fitness are reinitialized. To enable online adjustment of the search step size based on the current error, the convergence factor of IGWO is set as follows:

$$a=2\left(1-e^{-\left|error\right|}\right)·\left(1+rand\right)$$

(23)

where rand is a random number uniformly distributed in the range [0, 1]. error is the difference between the reference speed and the current actual speed.

5. Results and Discussion

To validate the effectiveness and superiority of the proposed IGWO-BPNN-FOPI control strategy, the controller presented in this paper is compared through simulations with several other control methods under identical parameters and environmental conditions. These methods include: the PI control method based on IGWO and BPNN (IGWO-BPNN-PI), ChASO-based FOPI (ChASO-FOPI) [17], and traditional GWO-based FOPI (GWO-FOPI) [16]. The parameters for IGWO, GWO, and ChASO are uniformly configured as shown in

Table 5. Algorithm parameter settings.

Parameter	Value
Population Size	20
Upper Bounds of Kp, Ki, and λ	[0.5 5 2]
Lower bounds of Kp, Ki, and λ	[0 0 0]

. The load torque of the PMSM is set to 10 Nm. The simulation step is set to 10⁻⁴ s, so the grey wolf’s position is updated every 10⁻⁴ s, and the controller parameters are updated accordingly.

The reference speed is set to 1500 r/min, and the speed step response curves of the PMSM under different control methods are shown in Figure 8. To intuitively compare the final control performance, several performance indicators are selected for comparison: response time (T_r), overshoot (M_p), settling time (T_s), and integral of time multiplied by squared error (ITSE). The calculation method of ITSE is shown in Equation (24). The simulation results are presented in

Table 6. Step response performance indicators for different control methods.

Control Method	Tr (s)	Mp (%)	Ts (s) (2%)	ITSE
IGWO-BPNN-FOPI	0.006	1.958	0.008	14.771
IGWO-BPNN-PI	0.007	5.065	0.052	17.102
ChASO-FOPI [17]	0.006	3.086	0.045	15.826
GWO-FOPI [16]	0.007	3.941	0.049	16.079

, from which it can be observed that the M_p, T_s, and ITSE of the PMSM under IGWO-BPNN-FOPI are superior to those of the other three methods. Especially for the T_s, it is reduced by 80% compared to other algorithms. This is primarily because the fitness function designed in this study incorporates multiple performance indicators rather than relying on a single metric for optimization. Additionally, owing to the BPNN’s accurate prediction of control effects, it ensures that the current controller parameters are optimal for the PMSM speed regulation system.

$$ITSE={\displaystyle {\int }_0^{+\infty }t{e}^2(t)dt}$$

(24)

To evaluate the response characteristics of PMSM under different control methods during continuous speed variations, the following rotational speed profile is specified: initial speed is 600 r/min, increasing to 1500 r/min at 0.2 s, then decreasing to 600 r/min at 0.4 s. These three speed changes correspond to three practical operational scenarios of PMSM: from zero to low speed, from low to high speed, and from high to low speed. The comparative response curves for continuous speed variations under different control methods are shown in Figure 9. It can be observed from the results that during continuous speed-variation processes, the IGWO-BPNN-FOPI control method demonstrates superior speed-tracking performance compared to other control approaches.

When the load changes, the speed also fluctuates. At 0.3 s, we reduced the load from 10 Nm to 3 Nm, and the resulting speed variation in the PMSM is shown in Figure 10. As shown in the figure, when the load decreases, the PMSM under IGWO-BPNN-FOPI control exhibits the smallest speed fluctuation. This illustrates that the method proposed in this paper possesses a certain degree of robustness.

To validate the load disturbance rejection performance of the IGWO-BPNN-FOPI control method, a 20 Nm load disturbance is added at 0.3 s in the simulation time. The load disturbance rejection effects of the PMSM under different control methods are shown in Figure 11 and Figure 12. Figure 11 shows the anti-load-disturbance performance of the PMSM at a speed of 600 r/min, while Figure 12 shows its performance at 1500 r/min. Whether at high or low speeds, the PMSM speed servo system using IGWO-BPNN-FOPI exhibits smaller speed variations and faster recovery to the original state after being subjected to the load disturbance. As can be seen from Figure 13 and Figure 14, this may be because the IGWO-BPNN-FOPI control method can adaptively adjust the search step size of IGWO based on real-time error, enabling rapid tuning of controller parameters. When speed fluctuation occurs and leads to increased instantaneous error, the IGWO will search for more suitable controller parameters, thereby quickly regulating the motor speed to the preset value.

The only difference between IGWO-BPNN-FOPI and IGWO-BPNN-PI lies in the use of a fractional-order versus an integer-order controller. However, across simulations of speed step response, continuous speed variation response, and load disturbance, IGWO-BPNN-FOPI consistently demonstrates the best performance, while the latter shows the poorest. This clearly indicates that fractional-order controllers possess distinct advantages for PMSM servo systems.

We have also considered the implementation of the proposed IGWO-BPNN-FOPI strategy on embedded systems. Since the strategy performs online parameter tuning, it imposes certain performance requirements on the embedded device. However, with the continuous advancement of electronic technology, the clock speed and memory capacity of microcontrollers have been steadily increasing, now capable of meeting our demands. For instance, mainstream ARM Cortex-M7 cores can achieve clock speeds of up to 550 MHz, fulfilling both the computational and real-time requirements of IGWO-BPNN-FOPI. Furthermore, microcontrollers based on this core can offer Flash memory of up to 1 MB, while the code and data footprint of this strategy does not exceed 50 KB, fully satisfying the storage needs for both programme and data.

In addition to the aforementioned performance requirements for embedded devices, the method proposed in this paper may have certain limitations. For example, the algorithm’s performance may exhibit some dependency on key parameters (such as the population size of IGWO), although we have identified a relatively optimal value. Moreover, the method is primarily suitable for small PMSMs under medium-to-low-speed conditions, such as the motor selected in Section 1 and the test speeds discussed later. Its online tuning capability is expected to demonstrate better dynamic performance compared to fixed-parameter controllers, particularly under dynamic operating conditions like sudden load changes.

In future work, we will further address the existing issues to make the method proposed in this paper applicable to a wider range of more complex scenarios.

6. Conclusions

To address the difficulty of designing FOPI controller parameters in PMSM systems, the improved grey wolf optimization algorithm and backpropagation neural network are applied to the fractional-order control of PMSM. The conclusions of this paper are summarized as follows:

(1)

This paper proposes an improved grey wolf optimizer. The Chebyshev chaotic mapping is introduced during the algorithm’s initialization phase to enhance individual diversity. A new nonlinear convergence factor is proposed, which improves the search accuracy in the later stages of the algorithm. A stochastic position update formula is introduced to avoid falling into local optima. The improved GWO is tested using six benchmark functions, and the results demonstrate that these modifications are effective. Furthermore, compared to several other algorithms, IGWO exhibits faster convergence speed and better stability.

(2)

The BPNNs integrated with IGWO accurately fit the relationship between FOPI controller parameters and control performance metrics. By employing IGWO to update the weights and biases of the BPNNs, the training effectiveness of the networks is enhanced. The BPNNs achieve high fitting accuracy, with an error of only 3% in fitting the overshoot.

(3)

This paper designs a FOPI control strategy capable of online parameter adjustment. A fitness function is developed based on the performance indicators approximated by the BPNNs, enabling real-time evaluation of current control parameters. Consequently, the IGWO achieves online tuning of the FOPI controller parameters. This FOPI control strategy is successfully applied to PMSM servo systems.

(4)

The simulation results of the speed step response show that under the IGWO-BPNN-FOPI control method, the PMSM exhibits an overshoot of 1.958%, a settling time of 0.008 s, and an ITSE of 14.701. The proposed method demonstrates better control performance compared to the other three methods. In the load disturbance simulation, the PMSM under the proposed method experiences the smallest speed variation and the fastest recovery after being subjected to a load disturbance, indicating its stronger resistance to such disturbances.

In future research, the IGWO-BPNN-FOPI control strategy will be implemented in embedded devices. Concurrently, this approach will be extended to more precise servo systems and complex nonlinear control systems.