Radial Basis Function Neural Network and Feedforward Active Disturbance Rejection Control of Permanent Magnet Synchronous Motor

Abstract

A composite control strategy is proposed to improve the position-tracking performance and anti-interference capabilities of permanent magnet synchronous motors (PMSMs). This strategy integrates an active disturbance rejection controller (ADRC) and a radial basis function neural network (RBFNN) with feedforward control. Initially, the flexibility and robustness of the ADRC are utilized in the position loop control. Subsequently, the parameters of the extended state observer (ESO) within the ADRC are optimized, benefiting from the fast convergence speed and optimal approximation provided by the RBFNN. To further enhance the dynamic tracking performance, a differential feedforward link is introduced between the desired speed and the output signal. The simulation and experimental results demonstrate that when the expected electrical angle inputs are sinusoidal and pulse signals, the incorporation of the feedforward link and the adjustment of parameters in the ADRC lead to improved position-tracking capabilities and greater adaptability to load disturbances.

1. Introduction

With the ongoing development of rare earth materials and advancements in power electronics technology, synchronous motors have gradually surpassed induction motors in terms of torque density and operating efficiency [1,2]. Among these, permanent magnet synchronous motors (PMSMs) are widely used in servo control applications due to their high efficiency, reliability, power density, and ease of control [3,4].

However, the traditional control methods for PMSMs often fall short due to their complex nature as high-order systems with significant coupling and numerous adjustable parameters. These methods typically exhibit slow response speeds, excessive overshoot, and suboptimal control performance, which fail to satisfy the accuracy and response demands of position servo systems. Consequently, improving the high-performance control of PMSMs has become a significant focus for researchers in motor control theory and related disciplines [5].

Active disturbance rejection control (ADRC) offers a promising solution by estimating both the internal and external disturbances within the system. This approach is characterized by a robust series configuration that demonstrates low dependence on precise modeling [6,7].

After more than 20 years of development, many scholars have provided summaries and researched the related thoughts, theories, and advancements in this field. Professor Han Jingqing introduced active disturbance rejection control (ADRC) through unique analyses of proportional–integral–derivative (PID) control [8]. He elaborated on the evolution from PID control to ADRC, highlighting the advantages of active disturbance rejection control [9]. Professor Gao Zhiqiang further discussed the characteristics of and development trends for ADRC, focusing specifically on disturbance suppression [10].

Active disturbance rejection control operates on the fundamental principle of PID deviation control, allowing it to function without the need for precise modeling of the control object. This approach effectively addresses the challenge of balancing rapid response with overshoot. ADRC is characterized by high control accuracy, fast response times, and strong stability. However, it includes numerous adjustable parameters that can lead to increased interdependency among them. It is important to recognize that the performance index of the state observer significantly affects both the precision and stability of the controller [11,12,13].

Numerous scholars have researched active disturbance rejection control (ADRC) and its enhancements to control strategies to address the challenge of disturbance suppression in permanent magnet synchronous motors. The fundamental issues associated with ADRC primarily encompass three areas: parameter tuning and optimization, performance analysis, and stability/convergence proof [14]. Furthermore, improvements to ADRC can be classified into two categories: one involves enhancing ADRC through the introduction of additional tools, while the other focuses on refining various components within the ADRC framework, such as the tracking differentiator (TD), the extended state observer (ESO), and the nonlinear state error feedback law (NLSEF) [14]. In reference [15], the ESO is improved based on error principles to enhance the nonlinear function. Reference [16] describes enhancing the ESO structure by integrating first-order active disturbance rejection, which facilitates better disturbance compensation in the system through the addition of supplementary terms. Additionally, reference [17] proposes a modified third-order extended state observer. The performance of the controller can be significantly improved by enhancing its tracking capabilities and disturbance rejection through the implementation of the proportional–integral disturbance update law. Reference [18] addresses the challenge of the measurement noise resulting from the bandwidth of the extended state observer (ESO) in an analysis of an enhanced active disturbance rejection controller. This analysis employs frequency-domain characteristics and utilizes finite-time technology. In reference [19], the parameters of the controller are optimized using an active disturbance rejection algorithm based on a backpropagation (BP) neural network. This optimization enhances the accuracy of the disturbance estimations from the extended state observer, leading to an improved dynamic performance and robustness of the control system. Reference [20] further incorporates an artificial intelligence algorithm into the parameter optimization of the active disturbance rejection controller, resulting in noticeable performance improvements. Currently, the optimization of the active disturbance rejection controller (ADRC) parameters primarily focuses on the bandwidth or the parameter combination. Among these methods, the latter presents distinct advantages and offers greater research opportunities [21].

In summary, various control strategies each have distinct characteristics, but they are not universally applicable. To enhance the control performance, the current trend is to develop composite control strategies that integrate other methods with active disturbance rejection control (ADRC) [22].

Table 1. Comparison of the advantages and disadvantages of various research methods.

	ADRC	RBF-ADRC	FRBF-ADRC
Advantages	It has strong robustness, a fast response speed, and good steady-state accuracy.	It has a strong adaptive adjustment ability, strong robustness, and a strong optimization control ability.	It has good dynamic performance, strong robustness, and a fast response speed.
Disadvantages	The implementation and calculation are more complicated.	The training data are more complex, and it is highly dependent on the training data.	The parameter tuning requirements are higher, and the system dependence is strong.

compares the characteristics of ADRC, RBF-ADRC, and feedforward–RBFNN-ADRC. Feedforward–RBFNN-ADRC demonstrates the most effective performance. Consequently, this paper will focus on the RBF neural network ADRC strategy with a feedforward approach, aiming to further improve the positional control.

This paper presents a compound control method that combines active disturbance rejection control (ADRC) with a radial basis function (RBF) neural network for position control of permanent magnet synchronous motors (PMSMs). This method harnesses the fast convergence and strong approximation capabilities of the RBF neural network. ADRC is integrated into the control strategy for the PMSM servo system. Specifically, a second-order nonlinear ADRC is employed to achieve combined control of both the position and speed, while the RBF neural network is utilized to compensate for and adjust the extended state observer within the controller. Additionally, to minimize dynamic tracking errors in terms of the system’s position, a differential feedforward link is introduced between the reference signal and the speed output. This approach simplifies the control structure to some extent and enhances the dynamic response speed.

2. The Mathematical Model

2.1. The Permanent Magnet Synchronous Motor Model

Taking a surface-mounted PMSM as an example, its angular velocity, mechanical motion, torque, and voltage equations under the d, q axis are as follows:

$$\{\begin{array}{c}{\displaystyle \frac{d\theta }{dt}}=\omega \\ J{\displaystyle \frac{d\omega }{dt}}=T_e-T_L-B\omega \\ T_e={\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q\\ u_d=R{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-\omega L_q{i}_q\\ u_q=R{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+\omega \left(L_d{i}_d+{\psi }_f\right)\end{array}$$

(1)

In this formula, θ is the mechanical angle of the rotor; ω is the mechanical angular velocity of the rotor; J is the rotational momentum of the motor; B is the friction coefficient; p_n is the pole-pair number of the permanent magnet synchronous motor; T_e is the electromagnetic torque; T_L is the load torque; ψ_f is the flux of the permanent magnet connecting rod; u_d and u_q are the quadrature axis voltage and the direct axis voltage; i_d and i_q are the orthogonal axis current and the DC axis current, respectively; and L_d and L_q are the orthogonal axis inductance and the straight axis inductance, respectively. R is the phase winding resistance.

The following results can be obtained by using the control method of i_d = 0:

$$\{\begin{array}{l}{u}_d=-\omega L_q{i}_q\hfill \\ u_q=R{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+\omega {\psi }_f\hfill \end{array}$$

(2)

Formula (2) is the voltage equation for the d, q axis when i_d = 0. i_d = 0 control is also known as d-axis constant current control. Combined with Formulas (1) and (2), it can be seen that the excitation component of the stator armature current part of the PMSM is always 0, and the equivalent d-axis open circuit has no effect. At this time, the torque equation is only proportional to the q-axis current. To adjust the torque and speed, it is only necessary to control i_q. On the other hand, when the q axis’s polarity changes, the torque direction will also change, so braking is relatively simple. The d-axis constant current control has a good dynamic response and torque regulation performance.

2.2. The Mathematical Model of ADRC

The active disturbance rejection control (ADRC) system integrates the classical proportional–integral–differential (PID) control concept based on error elimination and combines it with the advanced achievements of modern control theory. It usually consists of three components: a tracking differentiator (TD), an extended state observer (ESO), and the nonlinear error feedback control law (NLSEF). The TD helps to smooth the transition of the input command signal and alleviates the trade-off between speed and smoothness in differential operations. The core of ADRC is the ESO, which is responsible for estimating and compensating for internal and external disturbances. The error feedback control law mainly focuses on compensating for and adjusting the error between the TD and ESO outputs. The schematic diagram of the ADRC is as follows:

As shown in Figure 1, v is a given reference input, which outputs the tracking signal v₁ and the differential v₂ of the tracking signal through the TD. At the same time, the output y passes through the ESO to obtain the extended state z₁, z₂ of the controlled object, and then the overall disturbance z₃ of the system is obtained. Further, it compensates for the target error e₁ and the differential error e₂ in the system according to feedforward compensation. Finally, the difference between the outputs of the TD and the ESO is passed through the NLSEF and the output u₀, together with feedforward compensation. In addition, u₀ is the intermediate variable for the output, u is the final output, w is the unknown disturbance acting on the controlled object, b₀ is the gain, and 1/b₀ is the compensation coefficient.

Referring to the control structure in Figure 1, the second-order active disturbance rejection model can be derived as follows (it includes three parts: the TD, the NLSEF, and the ESO):

$$\mathrm{TD}:\{\begin{array}{l}fh=fhan(v_1(t)-v(t),v_2(t),r,h)\hfill \\ \begin{array}{l}{v}_1(t+1)=v_1(t)+h{v}_2(t)\\ v_2(t+1)=v_2(t)+hfh\end{array}\hfill \end{array}$$

(3)

h in Formula (3) above represents the integral step size. v(t) is the input signal, v₁(t) corresponds to the tracking signal of v(t), and v₂(t) denotes the estimated integral of v(t), fh represents the fastest control function fhan(v₁(t) − v(t), v₂(t), r, h).

$$\mathrm{The} \mathrm{ESO}:\{\begin{array}{l}e=z_1-y\hfill \\ \begin{array}{l}\stackrel{·}{z_1}=z_2-{\beta }_{01}fal\left(\epsilon ,{\alpha }_1,{\lambda }_1\right)\\ \stackrel{·}{z_2}=z_3-{\beta }_{02}fal\left(\epsilon ,{\alpha }_2,{\lambda }_2\right)+bu(t)\end{array}\hfill \\ \stackrel{·}{z_3}=-{\beta }_{03}fal\left(\epsilon ,{\alpha }_3,{\lambda }_3\right)\hfill \end{array}$$

(4)

Formula (4) includes the error estimate e, where y represents the output value, z₁ is the estimated value of y, z₂ is the derivative of z₁, and z₃ represents the estimated value of the internal and external disturbances in the system. Additionally, β₀₁, β₀₂, and β₀₃ denote the increments for correcting output errors, and λ₁, λ₂, and λ₃ are filter factors. Lastly, b is the coefficient for compensating for disturbances.

$$\mathrm{The} \mathrm{NLSEF}:\{\begin{array}{l}{e}_1=v_1-z_1\\ e_2=v_2-z_2\\ u_0={\beta }_{03}fal\left(\epsilon ,{\alpha }_3,{\lambda }_3\right)\\ u=u_0-z_3/b\end{array}$$

(5)

The formula above shows e₁ and e₂ as error signals, with fal denoting the nonlinear function. The expression is depicted as follows:

$$fal\left(e,\alpha ,\lambda \right)=\{\begin{array}{l}{\displaystyle \frac{e\left(t\right)}{{\lambda }^{\alpha -1}}},\left|e\left(t\right)\right|\le \lambda \hfill \\ {\left|e\left(t\right)\right|}^{\alpha }sign\left(e\left(t\right)\right),\left|e\left(t\right)\right|>\lambda \hfill \end{array}$$

(6)

The interval length λ in the formula represents the linear segment of the function, while α is the parameter for the nonlinear function where 0 < α < 1. sign(∙) denotes a symbolic function.

2.3. The Mathematical Model of the RBF Neural Network

The RBF neural network is an artificial neural network that closely resembles a multi-layer feedforward network. It features a three-layer architecture with a single hidden layer. Known for its exceptional local approximation capabilities, this network offers a straightforward and practical training approach. Its versatility has made it a popular choice for applications in system modeling, control, and various other fields. Refer to Figure 2 for a topology diagram of the network.

As shown in Figure 2, $I={\left[I_1,I_2,I_3\right]}^T$ is the input vector; $J={\left[J_1,J_2,J_3,J_4,J_5,J_6\right]}^T$ is the radial basis vector; and $O={\left[O_1,O_2,O_3\right]}^T$ is the weight vector.

The diagram illustrates that the neuron nodes representing the input signal constitute the input layer of the network’s first layer. The number of neurons in this layer is determined by the dimensions of the input data. The second layer is the hidden layer, which transforms the input layer’s data into a higher-dimensional space, facilitating the classification of different data types. The number of neurons in the hidden layer can be adjusted flexibly, and multiple neurons can be structured to address the system’s complexity. This design allows for both an effective approximation performance and real-time processing. The third layer is the output layer, which generates outputs based on weighted contributions from the hidden layer and typically consists of a single neuron output structure.

Moreover, the components labeled β₀₁, β₀₂, and β₀₃ in the network, along with the ESO nonlinear function factor in the active disturbance rejection controller, can be restructured simultaneously. In an attempt to improve the influence of the hidden layer’s excitation function, the Gaussian function is generally selected.

2.4. The Mathematical Basis of Feedback and Feedforward Control

In a feedback control system, the feedback object is controlled by deviation. When interference is present, the controlled variable initially strays from the desired value, prompting the regulator to carry out corrective action based on this deviation to counteract the interference. With continuous interference, the system consistently lags behind, and this ultimately results in a steady-state tracking error, impacting the system’s final trajectory and leading to machine errors.

Feedforward control operates as an open-loop control system that adjusts in response to disturbances; when a disturbance affects the system, corrective action is immediately triggered based on the level of the disturbance. In theory, feedforward control can effectively eliminate any deviations caused by disturbances. Because the positioning accuracy of the permanent magnet synchronous motor cannot meet the requirements of the high-precision servo system and the dynamic response speed of the position is slow, there is a large tracking error, and there is still much room for improvement. Therefore, a differential feedforward link is added between the given signal and the given speed output to improve the dynamic tracking performance for the system’s position and reduce the position dynamic tracking error to a certain extent [23].

Additionally, the primary role of feedforward control is to increase the system’s bandwidth, alleviate the controller’s workload, and decouple the nonlinear components. Most of the current control methods rely on differential adjustments. This means that when the system’s output deviates from its expected value, it adjusts based on that deviation. In contrast, feedforward control monitors the amplitude and behavior of the disturbance signals. The feedforward controller then responds correspondingly to compensate for these disturbances in advance. This proactive approach minimizes the impact of interference on the system’s output, thereby achieving a stable output.

Moreover, feedforward control enables direct control without delay, which enhances the system’s response rate. However, it requires an accurate understanding of the controlled object’s model and the system characteristics. Compared to feedback control, feedforward control offers more timely adjustments and is not hindered by system lag. The structure of feedforward control is illustrated in Figure 3.

The transfer function from the input X(s) to the system output Y(s) can be observed in the figure above.

$$G_0(s)={\displaystyle \frac{Y(s)}{X(s)}}={\displaystyle \frac{G_p(s)G_b(s)+G_f(s)G_b(s)}{1+G_p(s)G_b(s)}}$$

(7)

Moreover, in a system with an error E(s) = X(s) − Y(s), the transfer function from the system error to the input is

$$G(s)={\displaystyle \frac{E(s)}{X(s)}}={\displaystyle \frac{1-G_f(s)G_b(s)}{1+G_p(s)G_b(s)}}$$

(8)

If 1 − G_f(s)G_b(s) = 0, then G_f(s) = 1/G_b(s); then, the system error E(s) can be made zero.

In practical applications, achieving a system error of zero is impossible, but it is possible to minimize the tracking error within an acceptable range. When the system incorporates feedforward control versus not incorporating feedforward control, the denominator of the transfer function remains the same, indicating that the poles of both transfer functions are identical. Therefore, the introduction of feedforward control does not compromise the system’s stability; instead, it significantly enhances the system’s steady-state accuracy without altering the initial system’s parameters or structure. Furthermore, ensuring the dynamic performance of the system is also easily achieved with the implementation of feedforward control.

3. Improved Active Disturbance Rejection Control Strategy Design

The article presents a novel approach to position control for PMSMs using a combined control method that integrates ADRC with RBF neural networks. Initially, ADRC is employed to control the PMSM servo system. Furthermore, leveraging the fast convergence and optimal approximation properties of RBF neural networks, the parameters of the extended state observer (ESO) in ADRC are adjusted and fine-tuned. Additionally, to enhance the system’s dynamic tracking performance, a feedforward link is incorporated. The overall control block diagram is illustrated in Figure 4 to demonstrate these integrated control strategies.

The improved active disturbance rejection control structure still adopts PI control in the current loop and uses the vector control of i_d = 0 for the control object, the PMSM. The control principle can be roughly described as measuring the stator currents i_a, i_b, and i_c of the three-phase motor or only measuring any two phases and calculating the remaining one phase. The voltage components u_d and u_q of the d, q axis are obtained by Clarke and Park transformation and then adjusted by the ADRC + RBF neural network + feedforward control. Finally, the u_α and u_β components in the stationary coordinate system are obtained by inverse Park transformation. Then, the three-phase sinusoidal voltage is generated by SVPWM (Space Vector Pulse Width Modulation) to control the operation of the motor.

3.1. Design of an Auto Disturbance Rejection Controller

3.1.1. The TD Design

The second-order nonlinear tracking differentiator (TD) system is employed to extract the desired electrical angle signal and the other anticipated mutation signals. This approach allows for a well-organized transition process. The desired electrical angle signal can be effectively filtered, leading to the extraction of a noise-suppressed version of the signal. The discrete expression for this second-order nonlinear TD system is as follows:

$$\{\begin{array}{l}fh=fhan(x_1,x_2,r,h)\\ x_1(k+1)=x_1(k)+T{x}_2(k)\\ x_2(k+1)=x_2(k)+T\cdot fh\end{array}$$

(9)

In Formula (9), the state variables x₁ and x₂ represent the expected electric angle and the expected electric angle change rate (electric angular velocity), respectively; r is the speed factor, and its parameter is adjustable. When its value is larger, the expected command tracking speed is faster. h is the filtering factor, and the greater h, the better the filtering effect; T is the differential interval. The smaller T is, the better the filtering effect is. In general, T is slightly smaller than h.

The formula for the algorithm solving process in Equation (10) is

$$\{\begin{array}{l}d=r\cdot h\\ d_0=h\cdot d\\ \rho =x_1+h{x}_2-u(k)\\ a_0=\sqrt{(d^2+8r|\rho |)}\\ a=\{\begin{array}{l}{x}_2+(a_0-d)\cdot sign(\rho )/2, \left|\rho \right|>d_0\\ x_2+\rho /h, \left|\rho \right|\le d_0\end{array}\\ fhan(x_1,x_2,r,h)=\{\begin{array}{l}-r\cdot sign\left(a\right), \left|a\right|>d\\ -r\cdot a/d, \left|a\right|\le d\end{array}\end{array}$$

(10)

In Formula (11), u(k) is the input expected electrical angle signal; sign(∙) is a sign function.

3.1.2. Design of the Third-Order Nonlinear ESO

For second-order nonlinear systems, the corresponding equations are

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f(x_2,w,t)+bu\\ \rho =x_1\end{array}$$

(11)

In Formula (11), x₁ is the system state variable θ_e, and x₂ is the system state variable ${\dot{\theta }}_e$; $f(x_2,w,t)=f({\dot{\theta }}_e,T_L,t)+w$ contains unknown disturbances and known disturbances, which can be extended to a second-order system with the disturbance state variable $x_3=f(x_2)$. At the same time, let ${\dot{x}}_3=w_{rd}(t)$; in selecting the appropriate compensation factor b (b > 0), state variable $\widehat{x}$ of the state observer can track state variable x of the system.

Therefore, Formula (12) can be extended to

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=w_{rd}(t)+bu\\ {\dot{x}}_3=w_{rd}(t)\\ \rho =x_1\end{array}$$

(12)

The state observer of the system is constructed as follows:

$$\{\begin{array}{l}{e}_1=z_1-\rho \\ {\dot{z}}_1=z_2-{\beta }_{01}\mathrm{fal}(e_1,{\alpha }_1,{\lambda }_1)+b{i}_q^{*}\\ {\dot{z}}_2=z_3-{\beta }_{02}\mathrm{fal}(e_1,{\alpha }_1,{\lambda }_1)\\ {\dot{z}}_3=-{\beta }_{03}\mathrm{fal}(e_1,{\alpha }_2,{\lambda }_2)\end{array}$$

(13)

In Formula (13), ρ is the actual θ_e of the motor; z₁ is the observed value of θ_e. z₂ is the observed value of ${\dot{\theta }}_e$; e₁ is the observation error of θ_e; z₃ is the observed value of the total disturbance $f(x_2,w,t)$; $i_q^{*}$ is the control quantity of the ADRC’s output; β₀₁ is the output correction factor for observation error θ_e; β₀₂ is the output correction factor for observation error ${\dot{\theta }}_e$; β₀₃ is the output correction factor for the disturbance observation error.

The expression of the $fal(\epsilon ,\alpha ,\lambda )$ function is

$$fal(\epsilon ,\alpha ,\lambda )=\{\begin{array}{ll}|\epsilon {|}^{\alpha }\cdot \mathrm{sgn}(\epsilon ),& \hfill \left|\epsilon \right|>\lambda \\ \epsilon /{\lambda }^{1-\alpha },& \hfill \left|\epsilon \right|\le \lambda \end{array}$$

(14)

In the formula, ε is the error variable of the input; λ is the linear interval width of the fal function, which is related to the range of system error. α is the power of the exponential function. When the lumped disturbance $\tilde{d}=0$, the third-order nonlinear extended state observer is asymptotically stable near the zero point of the equilibrium point, and the differential equation of Equation (15) can be transformed into the following form:

$$\dot{\tilde{\epsilon }}=-\widehat{A}(\tilde{\epsilon })\tilde{\epsilon }=-\left[\begin{array}{ccc}{\beta }_1& -1& 0\\ {\beta }_2\vartheta & 0& -1\\ {\beta }_3\vartheta & 0& 0\end{array}\right]\left[\begin{array}{c}{\tilde{\epsilon }}_1\\ {\tilde{\epsilon }}_2\\ {\tilde{\epsilon }}_3\end{array}\right]$$

(15)

We reference Definition [24]. If there is a matrix D whose main diagonal elements are all positive so that the matrix $D\widehat{A}(\tilde{\epsilon })$ is symmetric positive definite, then the zero solution for the observer error equation of Equation (14) is Lyapunov asymptotically stable. Take matrix D as

$$D=\left[\begin{array}{ccc}{d}_{11}& d_{12}& d_{13}\\ -d_{12}& d_{22}& d_{23}\\ -d_{13}& -d_{23}& d_{33}\end{array}\right]$$

(16)

According to Formulas (15) and (16), we can obtain

$$\mathrm{D}\widehat{\mathrm{A}}(\tilde{\epsilon })=\left[\begin{array}{ccc}{D}_{11}& -d_{11}& d_{12}\\ D_{21}& d_{12}& -d_{22}\\ D_{31}& d_{13}& d_{23}\end{array}\right]$$

(17)

$$\{\begin{array}{c}{D}_{11}=d_{11}{\beta }_{01}+d_{12}{\beta }_{02}{\delta }_1+d_{13}{\beta }_3{\delta }_{01}\\ D_{21}=-d_{12}{\beta }_{01}+d_{22}{\beta }_{02}{\delta }_1+d_{23}{\beta }_3{\delta }_{01}\\ D_{31}=-d_{13}{\beta }_{01}-d_{23}{\beta }_{02}{\delta }_1+d_{33}{\beta }_3{\delta }_{01}\end{array}$$

(18)

To simplify the derivation of stability, under the condition that the main diagonal elements of matrix D are positive, the conditions are set as follows:

$$\{\begin{array}{c}{d}_{11}=1\\ d_{22}=d_{33}=\sigma \end{array}$$

(19)

Formula (19) is organized as follows:

$$d_{23}={\displaystyle \frac{1+\varpi {\beta }_{01}^2+\varpi {\beta }_{01}{\beta }_{03}\vartheta +\varpi {\beta }_{02}\vartheta }{\left({\beta }_{01}{\beta }_{02}-{\beta }_{03}\right)\vartheta }}$$

(20)

In the above formula, $\varpi$ is a positive number to be determined and is greater than 0, $\vartheta =fal({\tilde{\epsilon }}_1)/{\tilde{\epsilon }}_1$. It is known from Formula (20) above that when the gain coefficient of the observer satisfies the stability condition (β₀₁ β₀₂ − β₀₃) > 0, the symmetric matrix $D\widehat{A}(\tilde{\epsilon })$ is positive definite, and the zero solution of the matrix is Lyapunov asymptotically stable.

3.1.3. Design of the NLSEF

The traditional proportional–integral–differential (PID) control method struggles to fulfill the system’s performance requirements solely by summing up the errors from individual elements. It also faces challenges in balancing between the parameters of speed and overshoot. The nonlinear state error feedback (NLSEF) law replaces the linear combination form with a nonlinear structure. This enhances the information processing efficiency and improves the system’s performance to some extent. Additionally, dynamic compensation linearization is achieved through the compensation of the extended state observer. In the NLSEF, the system’s state error compares the output $z_i(i=1,2,\dots ,n)$ of the extended state observer with the output $v_i(i=1,2,\dots ,n)$ of the tracking differentiator to determine the error amount:

$${\epsilon }_i=v_i-z_i(i=1,2,\cdots n)$$

(21)

The θ_e tracking signal of the TD is different from the observation signal θ_e in the observer, and then the linear PID combination is changed into the form of a series controller. The corresponding form of the NLSEF is

$$\{\begin{array}{l}{e}_2=v_1-z_1\\ e_3=v_2-z_2\\ i_{q0}={\beta }_1\mathrm{fal}(e_2,{\alpha }_2,\delta )+{\beta }_2\mathrm{fal}(e_3,{\alpha }_3,\delta )\\ i_q^{*}=i_{q0}-1/b\cdot z_3\end{array}$$

(22)

In Formula (22), $i_{q0}$ is the pre-set current control quantity of the q axis; $i_q^{*}$ is the final control quantity after the q axis plus the feedforward disturbance.

3.2. ADRC Regulation Design after the RBF Neural Network

Active disturbance rejection control employs a nonlinear combination of error feedback to influence the input control variables, significantly impacting the controller’s performance. By applying an intelligent control tuning method to the parameters of the PID controller, we introduce a neural network into the active disturbance rejection controller. Consequently, we design an active disturbance rejection controller based on this neural network approach.

As illustrated in Figure 5, we utilize the radial basis function (RBF) neural network in this study due to its stability, reliability, and versatility. The three parameters (β₀₁, β₀₂, and β₀₃) in the extended state observer of the active disturbance rejection controller are optimized online, capitalizing on the RBF neural network’s performance characteristics, including its rapid convergence speed and superior approximation.

The radial basis function outputs the Gaussian function, which is expressed as follows:

$$J_j=\exp \left(-{\displaystyle \frac{{\Vert I-D_j\Vert }^2}{2{\mu }_j^2}}\right)j=1,2,\cdots n$$

(23)

In Formula (23), $I={[I_1,I_2,\cdots ,I_n]}^T$ is the input vector, and $D_j={[d_{j1},d_{j2},\cdots d_{jn}]}^T$ is the center vector of the j node (hidden neuron) in the network. $\Vert •\Vert$ denotes the distance between the input vector and the center vector and the distance between the input neuron and the Gaussian center vector in the hidden layer. ${\mu }_j$ is the basis width parameter of the j hidden neuron node. The smaller ${\mu }_j$ is, the smaller the radial basis width is, and the more selective the basis function is.

In the control of a permanent magnet synchronous motor, the input vector of the identification neural network is

$$I=[i_q(k),\theta (k),\theta (k-1)]$$

(24)

In addition, the performance index function is chosen to describe the size of the approximation error:

$$E(k)={\displaystyle \frac{1}{2}}{e}_m^2(k)={\displaystyle \frac{1}{2}}{[{\theta }_e(k)-{\theta }_r(k)]}^2$$

(25)

To speed up the convergence of E(k), a momentum term is introduced, α is the momentum factor, and η is the learning rate, ranging from 0 to 1. Additionally, the iterative algorithm calculates the output weight, the node center value, and the node base width parameter of the network using the gradient descent method, as outlined below:

• Network output weighting quantity

The network output weighting is

$$O_j\left(k\right)=O_j\left(k-1\right)+\eta (\rho -{\rho }_m)J_j+\alpha \left(O_j\left(k-1\right)-O_{ij}\left(k-2\right)\right)$$

(26)

Among these, ρ_m is the neural network’s output, and ρ is the actual rotation angle of the motor.

2.

Node base width parameters

The node base width parameter is

$$\{\begin{array}{l}\Delta {\mu }_j\left(k\right)=\eta [\rho (k)-{\rho }_m(k)]O_j{J}_j{\displaystyle \frac{{\Vert I-D_j\Vert }^2}{{\mu }_j^3}}\\ {\mu }_j\left(k\right)={\mu }_j\left(k-1\right)+\Delta {\mu }_j\left(k\right)+\alpha [{\mu }_j\left(k-1\right)-{\mu }_j\left(k-2\right)]\end{array}$$

(27)

3.

Node center value

The node center value is

$$\{\begin{array}{l}\Delta d_i\left(k\right)=\eta [\rho (k)-{\rho }_m(k)]O_j{\displaystyle \frac{I-D_j}{{\mu }_j^2}}\\ {\mu }_j\left(k\right)={\mu }_j\left(k-1\right)+\Delta {\mu }_j\left(k\right)+\alpha [{\mu }_j\left(k-1\right)-{\mu }_j\left(k-2\right)]\end{array}$$

(28)

The algorithm for the Jacobian matrix is

$${\displaystyle \frac{\partial \rho (k)}{\partial i_q(k)}}\approx {\displaystyle \frac{\partial {\rho }_m(k)}{\partial i_q(k)}}={\displaystyle \sum _{j=1}^m{O}_j{J}_j\frac{d_{ji}-I(1)}{{\mu }_j^2}}$$

(29)

The gradient descent method is employed to adjust the correction parameters β₀₁, β₀₂, and β₀₃ in the ESO, and the control error is defined as $e_{rr}(k)=\theta (k)-{\theta }_m(k)$, which are derr (the error differential), err (error), and dderr (the differential of the error differential), $ee=[d{e}_{rr},e_{rr},dd{e}_{rr}]$:

$$\{\begin{array}{l}ee(1)=e_{rr}(k)-e_{rr}(k-1)\\ ee(2)=e_{rr}(k)\\ ee(3)=e_{rr}(k)-2e_{rr}(k-1)+e_{rr}(k-2)\end{array}$$

(30)

The index function of the RBF for ESO controller tuning is

$$F(k)={\displaystyle \frac{1}{2}}{e}_{rr}^2(k)$$

(31)

The final control output target is

$$\{\begin{array}{l}{i}_q(k)=i_q(k-1)+\Delta i_q(k)\\ \Delta i_q(k)={\beta }_{01}ee(1)+{\beta }_{02}ee(2)+{\beta }_{03}ee(3)\end{array}$$

(32)

4.

Adaptive parameter adjustment

By using the gradient descent method, the change in β₀₁, β₀₂, and β₀₃ is adjusted as follows:

$$\{\begin{array}{l}\Delta {\beta }_{01}=-{\eta }_{01}{\displaystyle \frac{\partial F}{\partial {\beta }_{01}}}=-{\eta }_{01}{\displaystyle \frac{\partial F}{\partial \rho }}{\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}{\displaystyle \frac{\partial \Delta i_q}{\partial {\beta }_{01}}}={\eta }_{01}{e}_{rr}(k){\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}ee(1)\\ {\beta }_{01}(k)=\Delta {\beta }_{01}+{\beta }_{01}(k-1)\\ \Delta {\beta }_{02}=-{\eta }_{02}{\displaystyle \frac{\partial F}{\partial {\beta }_{02}}}=-{\eta }_{02}{\displaystyle \frac{\partial F}{\partial \rho }}{\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}{\displaystyle \frac{\partial \Delta i_q}{\partial {\beta }_{02}}}={\eta }_{02}{e}_{rr}(k){\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}ee(2)\\ {\beta }_{02}(k)=\Delta {\beta }_{02}+{\beta }_{02}(k-1)\\ \Delta {\beta }_{03}=-{\eta }_{03}{\displaystyle \frac{\partial F}{\partial {\beta }_{03}}}=-{\eta }_{03}{\displaystyle \frac{\partial F}{\partial \rho }}{\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}{\displaystyle \frac{\partial \Delta i_q}{\partial {\beta }_{03}}}={\eta }_{03}{e}_{rr}(k){\displaystyle \frac{\partial \rho }{\partial \Delta i_q}}ee(3)\\ {\beta }_{03}(k)=\Delta {\beta }_{03}+{\beta }_{03}(k-1)\end{array}$$

(33)

3.3. The Basis of the Existing Introduction of the Feedforward Compensation Link

In order to address the impact of the compensation disturbance on the output position error during the dynamic process, a differential feedforward link has been introduced in Figure 6. This link connects the input signal with the position output to enhance the system’s dynamic tracking performance concerning the position accuracy and to decrease the positional tracking error. The detailed design of the feedforward compensation is illustrated in Figure 7. In the diagram, F_v(s) is the feedforward compensation function, k_pp, k_veq is the proportional gain of the position loop, T_veq is the time constant, k_veq/T_veqs + 1 is the transfer function of the response time of the position loop, and 1/s is the integral link.

According to the schematic diagram in Figure 3 above, combined with Figure 6 and Figure 7, we can bring the position loop into the schematic diagram to obtain the closed-loop transfer function G_fp in the position loop control system after adding differential feedforward compensation. The specific expression is as follows:

$$G_{\mathrm{fp}}={\displaystyle \frac{\left[F_{\mathrm{v}}(s)+k_{\mathrm{pp}}\right]k_{\mathrm{veq}}}{s\left(T_{\mathrm{veq}}s+1\right)+k_{\mathrm{pp}}{k}_{\mathrm{veq}}}}$$

(34)

To achieve the desired tracking performance for the system’s position, the closed-loop transfer function of the position loop is set to G_fp = 1, and the transfer function of the feedforward control is derived accordingly; the details are as follows:

$$F_{\mathrm{v}}(s)={\displaystyle \frac{s\left(T_{\mathrm{veq}}s+1\right)}{k_{\mathrm{veq}}}}={\displaystyle \frac{1}{k_{\mathrm{veq}}}}s+{\displaystyle \frac{T_{\mathrm{veq}}}{k_{\mathrm{veq}}}}{s}^2$$

(35)

In summary, the primary purpose of the structure utilizing feedforward compensation is to enhance the estimation of the observed disturbance signal and to predict future disturbance trends. The estimated disturbance signal is then directly added to the input of the controller as a feedforward compensation signal, allowing for proactive compensation for the disturbance’s impact on the system output. Additionally, the control signal generated by the radial basis function (RBF) neural network, the adjusted parameters of the extended state observer (ESO), and the control law of the active disturbance rejection controller are combined to create the final control command. Furthermore, the feedforward compensation signal is incorporated into the control directive to improve the system’s resistance to interference and its position-tracking capability.

4. Simulation Results and Analysis

A simulation model is built in MATLABR2022a/Simulink to verify the effectiveness of the method. The relevant motor parameters are shown in

Table 2. Parameters of the motor.

Argument	Value/Unit
Resistance Rs	2.6 Ω
Inductance Ld, Lq	0.00736 H
Moment of inertia J	0.00102 kg·m2
Flux linkage ψf	0.246 wb
Number of poles np	2
Polar distance τ	0.012 m
Quality m	3 kg

.

The existing loop employs a PI controller with the parameters set as follows: K_p = 100, K_i = 100. The position loop responses under ADRC and RBF-ADRC are illustrated in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, and the simulation analysis is categorized into four main sections, as described in Section 4.1, Section 4.2, Section 4.3 and Section 4.4.

4.1. The No-Load Condition of a Given 1 Rad Step Signal

When a 1 rad step signal is applied, the position curve is illustrated in Figure 8. Analyzing the curves in the graphs reveals that the controller reaches the specified position in approximately 0.0429 s despite there being noticeable jitter here, and the continuity is not optimal. In contrast, the RBF-ADRC controller can achieve the desired position in about 0.0339 s, providing a speed improvement of roughly 0.01 s over the original. Furthermore, FRBF-ADRC outperforms the previous controllers, reaching the specified position in approximately 0.0189 s, resulting in an additional speed enhancement of around 0.02 s.

Figure 8. Step signal position curve.

Figure 8. Step signal position curve.

4.2. The Loading Condition of a Given 1 Rad Step Signal

Figure 9 displays the position curve of a step signal subjected to a sudden load. When a sinusoidal signal at 1 rad is abruptly loaded with a 10 N force at 0.1 s, it becomes evident that conventional ADRC exhibits weak robustness. In contrast, RBF-ADRC shows rapid convergence to the target position curve, highlighting its superior robustness. Additionally, the performance of FRBF-ADRC is more effective than that of RBF-ADRC.

Figure 9. Step signal sudden load position curve.

Figure 9. Step signal sudden load position curve.

4.3. Controller Simulation under Different Step Signals

Figure 10 illustrates the controller position curves for different step signals. The position curves for the ADRC, RBF-ADRC, and FRBF-ADRC controllers are displayed for step signals of 1 rad, 1.5 rad, and 2 rad, highlighting the speed of each controller at various times. Utilizing the data from Figure 10, we can derive the comparative data presented in

Table 3. A comparison of simulations for ADRC, RBF-ADRC, and FRBF-ADRC controllers under different step signals.

Step Signal	ADRC	RBF-ADRC	FRBF-ADRC
1 rad	0.0429 s	0.0339 s	0.0189 s
1.5 rad	0.0553 s	0.0477 s	0.0434 s
2 rad	0.0632 s	0.0534 s	0.0515 s

. Among the three controllers, FRBF-ADRC demonstrates the best control performance. Additionally, as the position increases, the response time of the controller usually slows down.

Figure 10. The controller position curves under different step signals in the simulation.

Figure 10. The controller position curves under different step signals in the simulation.

4.4. Given a Sinusoidal Signal (Simulation)

Figure 11 presents the position and error curves of the controller, while Figure 12 is an amplification diagram of the sinusoidal position signal x = sin4πt. Analyzing the curves in these graphs reveals that under similar periodic sinusoidal changes within a specific range, the maximum absolute error for ADRC is |E_max| = 0.0196 rad, while the maximum absolute error for RBF-ADRC is |E_max| = 0.0173 rad. In contrast, the maximum absolute error for FRBF-ADRC is significantly lower, at |E_max| = 0.0043 rad. Therefore, it can be concluded that the FRBF-ADRC controller outperforms the other two controllers in terms of the absolute error.

Figure 11. Position and error curves of the controller.

Figure 11. Position and error curves of the controller.

Figure 12. Step signal position curve amplification.

Figure 12. Step signal position curve amplification.

5. Experimental Results and Analysis

In the comprehensive simulation, the results are validated using the RTunit platform (The software system version is RTunit Stido2020). The experimental validation setup is shown in Figure 13 below. The parameters used in both the experiment and the Simulink simulation are identical.

The hardware controller for the experiment is illustrated in Figure 14. To implement the controller and feedback system, either RTU-BOX204 or RTU-BOX201 (hardware controller) (RTunit Information Technology Co., Ltd., Nanjing, China) can be utilized. The actuator can be selected from the RTM-PEH series module or the RTI series three-phase driver, with the controlled object being a permanent magnet synchronous motor. Additional components of the hardware system include RTM-RACK (an integrated module including RTM-PEH series, etc.), RTC-Encoder (an encoder adapter board), RTC-ADC (an ADC adapter board), RTC-Fiber 6/12/24 (a fiber adapter board), and RTC-Power (a power adapter board) (RTunit Information Technology Co., Ltd., Nanjing, China). Furthermore, the encoder provides U, V, and W signals, which exhibit a phase difference of 120 degrees from each other. This phase relationship aids in estimating the initial position of the rotor’s d axis when the motor starts, enabling quick determination of the rotor’s approximate position during the initial phase.

It is important to note that the system employs a closed-loop feedback mechanism to keep the rotor in the designated position. Sensors, such as encoders, measure the rotor’s actual position and compare it to the set position. The resulting position error signal serves as the input for the controller. Based on this error signal, the controller adjusts the current components along the d axis and the q axis to generate the required torque for correcting the position error.

The results of the experiment are illustrated in Figure 15, Figure 16, Figure 17 and Figure 18. In alignment with the simulation analysis, the experimental validation is primarily categorized into four forms, as detailed in Section 5.1, Section 5.2, Section 5.3 and Section 5.4.

5.1. The No-Load Condition with a Given 1 Rad Step Signal

In Figure 15, it can be observed that when a 1 rad step signal is applied with the ADRC controller, the position aligns with the target curve at 0.1124 s. In contrast, the position reaches the desired curve at 0.0369 s when using the RBF-ADRC controller. The RBF-ADRC controller demonstrates enhanced speed and reduced chattering compared to the ADRC controller. Moreover, FRBF-ADRC is even more effective, achieving the desired curve at 0.0112 s with superior results.

Figure 15. Step signal position curve in the experiment.

Figure 15. Step signal position curve in the experiment.

5.2. The Loading Conditions of a Given 1 Rad Step Signal

Figure 16 illustrates that the ADRC controller demonstrates poor convergence when a 10 N load is suddenly applied within 0.1 s during the experiment. In contrast, the RBF-ADRC controller shows improved convergence. Notably, the FRBF-ADRC controller surpasses both the ADRC and RBF-ADRC controllers in terms of its convergence performance.

Figure 16. Curve diagram of step signal sudden loading position in experiment.

Figure 16. Curve diagram of step signal sudden loading position in experiment.

5.3. Experiments with the Controller under Different Step Signals

Figure 17 presents the position curves of the controller for various step signals based on experimental data. The curves for the ADRC, RBF-ADRC, and FRBF-ADRC controllers under 1 rad, 1.5 rad, and 2 rad step signals illustrate the response speed of each controller at specific times. Additionally, the comparison data displayed in

Table 4. Experimental comparison of ADRC, RBF-ADRC, and FRBF-ADRC controllers under different step signals.

Step Signal	ADRC	RBF-ADRC	FRBF-ADRC
1 rad	0.1124 s	0.0369 s	0.0112 s
1.5 rad	0.1429 s	0.0459 s	0.0446 s
2 rad	0.1545 s	0.0547 s	0.0526 s

can be derived from Figure 17. The FRBF-ADRC controller demonstrates a superior performance compared to the other two controllers. Furthermore, for each controller, a higher position correlates with a slower response time.

Figure 17. The controller position curves under different step signals in the experiment.

Figure 17. The controller position curves under different step signals in the experiment.

5.4. Given a Sinusoidal Signal (Experiment)

In Figure 18, when the sinusoidal signal x = sin4πt is applied, the ADRC controller has a maximum absolute error of |E_max| = 1.11 rad, RBF-ADRC has a maximum absolute error of |E_max| = 0.34 rad, and FRBF-ADRC has a maximum absolute error of |E_max| = 0.0022 rad. In conclusion, the error of the FRBF-ADRC controller is the lowest among the three.

Figure 18. Step signal position curve in the experiment.

Figure 18. Step signal position curve in the experiment.

6. Conclusions

This paper presents the application of active disturbance rejection control (ADRC) to the control of permanent magnet synchronous motors (PMSMs). It proposes a compound control method that integrates active disturbance rejection with radial basis function (RBF) neural networks and feedforward control. The active disturbance rejection controller is incorporated into the servo system of the PMSM, achieving position control through a second-order nonlinear active disturbance rejection controller. Additionally, the radial basis neural network adaptively compensates for and adjusts the extended state observer within the controller, enhancing the system’s capability to withstand load disturbances and improving the positioning accuracy. The simulation and experimental results show that the RBF-ADRC controller outperforms the conventional ADRC controller in terms of positioning, load convergence, and error management. The performance comparison tables (

Table 5. A comparison of simulations for ADRC, RBF-ADRC, and FRBF-ADRC controllers.

Signal Setting	ADRC	RBF-ADRC	FRBF-ADRC
Step signal rapidity	0.0429 s	0.0339 s	0.0189 s
Sudden load convergence	poor	mezzo	good
Given the sinusoidal signal error |Emax|	0.0196 rad	0.0173 rad	0.0043 rad

and

Table 6. Experimental comparison of ADRC, RBF-ADRC, and FRBF-ADRC controllers.

Signal Setting	ADRC	RBF-ADRC	FRBF-ADRC
Step signal rapidity	0.1124 s	0.0369 s	0.0112 s
Sudden load convergence	poor	medium	good
Given sinusoidal signal error |Emax|	1.11 rad	0.34 rad	0.0022 rad

) clearly illustrate these differences. Furthermore, the FRBF-ADRC controller surpasses the RBF-ADRC controller in terms of its overall performance.

This study has successfully achieved high precision and stability in controlling a permanent magnet synchronous motor by integrating a radial basis function (RBF) neural network with feedforward control and an active disturbance rejection control (ADRC) strategy. Notably, the FRBF-ADRC controller enhances the motor’s ability to resist interference and improves its dynamic response speed under disturbances, such as load changes and fluctuations in the position curve.

Secondly, the performance of the neural network parameter adjustment involved in this paper is affected by the training data. If the training data are not sufficient or there is a deviation, this may lead to a decrease in the control performance. In addition, there is still a lot of room for improvement in the parameter tuning of the active disturbance rejection controller and its immunity under multiple operating conditions.

Finally, because of the limitations of this study, the following directions are worthy of more in-depth study, as follows:

• To improve the learning ability and generalization ability of neural network controllers, the structure and parameters of the RBF neural network are worth further study and optimization, or a more advanced neural network could be used for the research;
• In terms of the improvement in the auto disturbance rejection controller, an auto disturbance rejection controller with linear and nonlinear switching could be introduced, combining a switching controller, the neural network method, and feedforward control to maximize the advantages of auto disturbance rejection;
• The active disturbance rejection controller could be combined with advanced methods other than neural networks from the perspective of a composite control strategy, such as active disturbance rejection control and deep learning, particle swarm optimization, etc.