Improved Active Disturbance Rejection Control for Permanent Magnet Synchronous Motor

Abstract

To improve the control performance of a permanent magnet synchronous motor (PMSM) under external disturbances, an improved active disturbance rejection control (IADRC) algorithm is proposed. Since the nonlinear function in the conventional ADRC algorithm is not smooth enough at the breakpoints, which directly affects the control performance, an innovative nonlinear function is proposed to effectively improve the convergence and stability. On this basis, the proposed IADRC is constructed, and comparative simulation results with ADRC and other IADRC show that faster response speed, higher accuracy and stronger robustness are obtained.

1. Introduction

Due to the advantages of high power density, simple structure, easy maintenance and convenient speed regulation, permanent magnet synchronous motors (PMSMs) are widely used in new energy vehicles, computerized numerical control (CNC) machine tools and other high-end equipment [1,2]. The proportional integral differential (PID) control technology, which is extensively utilized across various industries, governs the traditional PMSM speed regulation system. Although the PID controller has the advantages of simple structure and easy implementation [3,4], it is necessary to establish an accurate mathematical model of the controlled object to achieve accurate control. However, when modeling PMSMs, the motor structure is often simplified. This simplification has been proved inadequate when the PMSM operates at low speeds, resulting in a notable degradation of control performance, as highlighted in Ref. [5]. Moreover, noise and interference cannot be suppressed well by PID, so scholars have proposed a variety of improved PID systems [6,7]. By optimizing the adjustment parameters, the control performance can be significantly improved, but the anti-interference ability, response speed and accuracy still need to be enhanced.

Modern control theory provides a new solution for the performance improvement of control systems. Concurrently with the advancement in PMSM control technology, various effective control methods have been proposed. According to Ref. [8], the nonlinear PID controller enhances control performance through adaptive error signal transformation and dynamic adjustment of PID parameters, which enables the controller to efficiently attenuate noise in the input signal and address a trade-off between response speed and overshoot inherent in conventional PID controllers. Moreover, fuzzy PID is also adopted to enhance the system’s robustness [9]. Additionally, sliding mode control (SMC) has undoubtedly emerged as a critical advancement in motor control engineering in recent years [10,11]. It can enhance the control effectiveness of PMSM systems while exhibiting rapid convergence and strong resistance to interference. However, due to the chattering characteristics of sliding mode variable structure, how to effectively suppress this chattering has become an important topic. Robust control is also a class of methods aimed at uncertain systems, and robust control laws based on H_∞ paradigms and µ-synthesis have been studied, which achieve good disturbance suppression and robustness to parameter variations [12]. Although these algorithms effectively improve the PMSM system’s control effect, they still struggle with achieving superior control performance when addressing uncertainties in models and dealing with external interferences.

Active disturbance rejection control (ADRC) is a nonlinear control strategy proposed by Jingqing Han, which can effectively deal with uncertainties and external disturbances [13]. The utility of this approach is reflected in its superior anti-interference ability, strong adaptability to the change and uncertainty of system parameters, high control precision, fast response speed, simple structure and easy implementation [14], and it has attracted wide attention in many technical fields [15,16,17]. ADRC consists primarily of three components: tracking differentiator (TD), extended state observer (ESO), as well as nonlinear state error feedback control law (NLSEF). The TD can smooth the input signal, reduce the noise influence and provide differential information of the signal to help the controller better respond to changes in the system. The ESO is ADRC’s core component, which is used to estimate the state variables and total disturbances of the system. It has the capability to observe and mitigate uncertainties and external disturbances in real time, thereby enhancing the system’s robustness and resistance to interference. Additionally, the NLSEF generates control commands based on observed state error as well as disturbance information to obtain the controlled object’s accurate control. The nonlinear feedback control strategy provided by the NLSEF can dramatically enhance the system’s dynamic response. ADRC inherits the essence of traditional PID and can effectively control systems with nonlinearity and uncertainty without relying on accurate models [18,19]. For example, based on ADRC and sliding mode control, Fang et al. [20] proposed a new integrated design method of speed and position loops and realized the speed and position control of PMSMs. While achieving minimal system overshoot, the complexity of parameter configuration remains a challenge. Addressing this, Li et al. [21] introduced a sliding mode ADRC, replacing the ESO with a nonlinear disturbance observer to streamline parameter adjustments. Building upon their research, Ge et al. [22] employed particle swarm optimization algorithm to enhance ADRC for spacecraft attitude control. Instead of depending upon empirical selection, the parameters are now determined via mathematical optimization. Ref. [23] proposed a fractional-order fuzzy ADRC, which mitigates the limitations of conventional control methods (such as low precision and slow response), and it is used to manage the manipulator’s numerous joint motion control. Ramlavi and Chidan [24] studied the linear active disturbance rejection control tracking control approach to enhance the controller’s tracking performance and robustness and solved the model uncertainty and environmental disturbance of a single-wheeled robot. Drawing on Partovibakhsh and Liu’s research [25], Guo and Zhao [26] enhanced the tracking capabilities of mobile robots through the compensation mechanism of ADRC. Lv [27] proposes a fuzzy auto disturbance rejection control (Fuzzy-ADRC) method for a three-phase four-arm inverter for suppressing motor torque pulsations under complex operating conditions. Wang [28], in response to the poor control performance caused by fixed parameters in ADRC for a bearingless PMSM, proposes a dynamic parameter adjustment method for ADRC based on a genetic algorithm and backpropagation neural network. Fang [29] proposed an ADRC method based on an improved ESO to design an electromechanical actuator cascade controller for PMSMs.

Although ADRC technology effectively improves the control effect of nonlinear and uncertain systems, there are still limitations, such as the insufficiency of effective parameter-setting methodologies, the lack of estimation ability for fast time-varying disturbances and the fal function adopted by traditional ADRC not being smooth enough at the switching points between the nonlinear section and the linear section [30,31]. Considering the challenges currently faced in PMSM control systems with ADRC controllers, this paper aims to make the following contribution: a polynomial nonlinear function combined with a cosine function is proposed to improve the fal function, which effectively improves the nonlinear function’s smoothness and enhances ADRC’s robustness. Investigated findings indicate that when contrasting with the conventional ADRC and other improved ADRC (IADRC), the IADRC proposed herein significantly amplifies the system’s response speed, precision and robustness.

The organizational structure of this paper is outlined below. Section 2 introduces the PMSM’s mathematical model. Section 3 presents ADRC’s improvement strategy and constructs an IADRC controller. In Section 4, a design for the PMSM speed control system employing the proposed IADRC is presented. In Section 5, the simulation experiment and analysis are carried out, and the good control performance of the proposed IADRC for PMSM systems is verified.

2. Mathematical Modelling of PMSM

The rotor structure and permanent magnet distribution of the PMSM [32] are shown in Figure 1. According to their structural characteristics, PMSMs can be divided into two types: surface-mounted and built-in. The surface-mounted PMSM is also called the nonsalient pole PMSM. Because its permeability is very close to the vacuum permeability, the change between the reluctance and the inductance is very small, which makes the rotor have good nonsalient pole characteristics. This also makes the magnetic field of the permanent magnet have an approximately sinusoidal distribution, and the motor has a better performance. For the built-in PMSM, the reluctance of the direct axis is much higher than that of the quadrature axis, so the inductance of the direct axis is much lower than that of the quadrature axis and the rotor has salient pole characteristics, so it is also called the salient pole PMSM. The built-in PMSM realizes sensorless control through the salient pole effect and increases the power density through the electromagnetic resistance torque. However, the rotor structure is complex, the magnetic flux leakage coefficient is large and the manufacturing cost is high.

Due to the different structures of PMSMs, their operation modes and the formations of their electromagnetic torques will also be different. Therefore, to achieve a better control effect, it is necessary to formulate a variety of different control schemes according to the internal structures of PMSMs. Considering that the surface-mounted PMSM has low cost and the same inductance of the d-q axis, it is simpler to establish the electromagnetic torque equation. Therefore, the surface-mounted PMSM is selected in this study.

To analyze the three-phase PMSM’s mathematical model, the following assumptions are made for the convenience of analysis:

(1)

Ignore the reluctance of the stator and rotor cores, without considering losses due to eddy currents or hysteresis.

(2)

The permanent magnet material is nonconductive, and its permeability is equivalent to that of air.

(3)

The rotor does not have any damping windings.

(4)

The excitation magnetic field from the permanent magnet and the armature reaction magnetic field from the three-phase winding are sinusoidally distributed across the air gap.

(5)

The induced electromotive force waveform in the phase winding follows a sinusoidal pattern.

Assuming that the aforementioned conditions are met, according to electromagnetic induction and Kirchhoff’s voltage law, the voltage equation of a three-phase PMSM in the natural coordinate system can be expressed as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{a}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{a}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{a}}\\ u_{\mathrm{b}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{b}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{s}}\\ u_{\mathrm{c}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{c}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{c}}\end{array}\right.$$

(1)

The PMSM’s stator voltage equation in a two-phase rotating coordinate system can be obtained by the Clarke transform and Park transform [33]:

$$\left\{\begin{array}{l}{u}_{\mathrm{d}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{d}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{d}}-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{q}}\\ u_{\mathrm{q}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{q}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{q}}+{\omega }_{\mathrm{e}}{\psi }_{\mathrm{d}}\end{array}\right.$$

(2)

The stator magnetism-chain equation is:

$$\left\{\begin{array}{l}{\psi }_{\mathrm{d}}=L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}}\\ {\psi }_{\mathrm{q}}=L_{\mathrm{q}}{i}_{\mathrm{q}}\end{array}\right.$$

(3)

Bringing Equation (3) into (2) gives:

$$\left\{\begin{array}{l}{u}_{\mathrm{d}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{d}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{d}}-{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}\\ u_{\mathrm{q}}={\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{q}}}{\mathrm{d}t}}+R_{\mathrm{s}}{i}_{\mathrm{q}}+{\omega }_{\mathrm{e}}(L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}})\end{array}\right.$$

(4)

where $u_{\mathrm{a}},u_{\mathrm{b}},u_{\mathrm{c}}$ are the stator winding’s three-phase voltages, with V as the unit; $R_{\mathrm{s}}$ is the stator resistance, with Ω as the unit; $i_{\mathrm{a}},i_{\mathrm{b}},i_{\mathrm{c}}$ are the stator winding’s three-phase current, with A as the unit; ${\psi }_{\mathrm{a}},{\psi }_{\mathrm{b}},{\psi }_{\mathrm{c}}$ are the flux of the stator winding, with Wb as the unit; ${\psi }_f$ is the permanent magnet flux linkage, with Wb as the unit; $L_{\mathrm{d}},L_{\mathrm{q}}$ are the inductance components of the stator inductance on the dq axis, respectively, with H as the unit; ${\omega }_e$ represents the electrical angular velocity of the motor, with rad/s as the unit.

Through analyzing the components of resistance torque and dynamic torque, the motor’s mechanical motion equation is derived:

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}=T_{\mathrm{e}}-B{\omega }_{\mathrm{m}}-T$$

(5)

where $J$ is the moment of inertia, $T$ is the load torque and $B$ is the viscous friction coefficient. By Clarke transformation and Park transformation, the equation of electromagnetic torque is obtained below:

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}{p}_n\left[{\psi }_{\mathrm{f}}{i}_{\mathrm{q}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}}{i}_{\mathrm{q}}\right]$$

(6)

where $T_e$ is the electromagnetic torque; 3/2 is the coefficient when the equal amplitude transformation principle is adopted; $P_n$ pertains to the total number of motor pole pairs.

The transformed mechanical motion equation is:

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}={\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}\left[{\psi }_{\mathrm{f}}{i}_{\mathrm{q}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}}{i}_{\mathrm{q}}\right]-B{\omega }_{\mathrm{m}}-T$$

(7)

3. Design of Improved ADRC Controller

The traditional ADRC treats uncertainty, internal system disturbances, and external disturbances collectively as the system’s overall disturbance. And it can achieve real-time accurate estimation and compensation of this disturbance using the ESO and NLSEF. This approach does not rely on an exact model of the plant and significantly enhances anti-interference capabilities. However, the traditional ADRC’s fal function is not smooth enough at the switching points between the nonlinear section and the linear section, resulting in the control effect not being ideal. Therefore, an improved ADRC controller underpinned by an improved fal function is proposed in this paper.

3.1. Improved Fal Function Design

The fal function is a nonlinear function, which plays a vital role in ADRC. It not only allows ADRC to better estimate and compensate for system disturbances, thereby improving the system’s anti-disturbance performance, but also allows the amplification to be mitigated when the error is large and increased when the error is small. It is beneficial to enhance the system’s reaction velocity to large errors and the control accuracy to small errors, while enhancing the robustness and adaptability of the system.

In traditional ADRC, fal(e, α, δ), as a crucial nonlinear function, can effectively estimate both internal and external disturbances of the system, thereby generating the ADRC’s output signal. The form of the fal function is shown in Equation (8):

$$fal(e,\alpha ,\beta )=\left\{\begin{array}{l}{\left|e\right|}^{\alpha }sign(e),\left|e\right|\ge \delta \\ {\displaystyle \frac{e}{{\delta }^{1-\alpha }}}\begin{array}{cc}& \end{array},\left|e\right|<\delta \end{array}\right.$$

(8)

Taking δ = 0.001, when α takes different values, the curve of the fal function is shown in Figure 2.

From the perspective of convergence, the terminal attractor ${\left|e\right|}^{\alpha }sign(e)$ is finite-time convergent, especially suitable for the control near the origin stage (|e| < 1), which has an amplification effect on the error, but not suitable for the control away from the origin stage, and the convergence will be slower than the classical linear control. The linear term is exponentially convergent, which is suitable for the control far from the origin (|e| > 1). However, in the fal function, the use of the terminal attractor term and the linear term is reversed, which causes the fal function as a whole to converge in nonfinite time, no matter how small the initial error e(0) is, so the linear term can be improved to enhance the convergence ability.

From Equation (8), it is evident that the derivative values at the breakpoints differ after applying the derivative of the fal function. And it can also be seen from Figure 2 that the fal(e, α, δ) function curves are continuous but not smooth when e = |δ|, and there is a sudden change, which will lead to poor control performance of the system.

Based on the above problems, a polynomial nonlinear function Ifal(e, α, δ) combining linear and trigonometric functions is proposed in this paper. The reasons for adopting a combination of trigonometric functions and polynomials are as follows:

(1)

Trigonometric functions and polynomials have explicit mathematical forms, making them easy to handle analytically and theoretically.

(2)

Properly designed trigonometric functions and polynomials can reduce system overshoot and enhance the response speed and stability of the system.

(3)

Although the added trigonometric function makes Ifal more complex than the original fal function, the calculation efficiency can be improved by the look-up table and interpolation methods, and a better control effect can be achieved on the premise of ensuring real-time performance and lower hardware requirements.

The function expression is shown below:

$$Ifal(e,\alpha ,\beta )=\left\{\begin{array}{l}{\left|e\right|}^{\alpha }sign(e)\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}& \end{array}& & \end{array}& ,\left|e\right|\ge \delta \end{array}\\ {\displaystyle \frac{{\delta }^{\alpha -3}{e}^{k_1}\cos (e-\delta )}{k_1}}-{\displaystyle \frac{{\delta }^{\alpha -2}{e}^{k_2}\cos (e-\delta )}{k_2}}+k_3\delta +\alpha {\delta }^{\alpha -1}e,\left|e\right|<\delta \end{array}\right.$$

(9)

The design idea of the function is as follows: considering that the main problem of the original fal function is that the segments are not smooth enough, the original linear function is changed into a polynomial function combined with a trigonometric function and linear function when |e| ≤ δ in this paper. Then, according to the condition that the fal function is continuous at the segments and the derivative function values are equal, the relationship between the three constant coefficients of k₁, k₂ and k₃ is obtained.

Based on the above ideas, it can be known that when e = |δ|, the function values on both sides of the Ifal function are equal, and after taking the derivative of the Ifal function, the derivative values on both sides are also equal.

In light of the aforementioned concepts, the following equation is established:

$$\left\{\begin{array}{l}Ifa{l}_{-}(e,\alpha ,\beta )=Ifa{l}_{+}(e,\alpha ,\beta )\\ Ifa{l^{\prime }}_{-}(e,\alpha ,\beta )=Ifa{l}_{+}^{\prime }(e,\alpha ,\beta )\end{array}\right.$$

(10)

The derivative expression of the Ifal function, after calculation, is as follows:

$$Ifa{l}^{\prime }(e,\alpha ,\beta )=\left\{\begin{array}{l}\alpha {\delta }^{(\alpha -1)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ,\left|e\right|>\delta \\ {\delta }^{\alpha -3}{e}^{(k_1-1)}-{\delta }^{\alpha -2}{e}^{(k_2-1)}+\alpha {\delta }^{(\alpha -1)}\hspace{1em}\hspace{1em},\left|e\right|\le \delta \end{array}\right.$$

(11)

Combining Equations (10) and (11), we can derive Equation (12).

$$\left\{\begin{array}{l}{\displaystyle \frac{{\delta }^{\alpha -3}{e}^{k_1}\cos (e-\delta )}{k_1}}-{\displaystyle \frac{{\delta }^{\alpha -2}{e}^{k_2}\cos (e-\delta )}{k_2}}+k_3\delta +\alpha {\delta }^{\alpha -1}e={\left|e\right|}^{\alpha }sign(e)\\ {\delta }^{\alpha -3}{e}^{(k_1-1)}-{\delta }^{\alpha -2}{e}^{(k_2-1)}+\alpha {\delta }^{(\alpha -1)}=\alpha {\delta }^{(\alpha -1)}\end{array}\right.$$

(12)

When |e| = δ, Equation (3) simplifies to the following form:

$$\left\{\begin{array}{l}{\displaystyle \frac{{\delta }^{k_1+\alpha -3}}{k_1}}-{\displaystyle \frac{{\delta }^{k_2+\alpha -2}}{k_2}}+k_3\delta +\alpha \delta ={\delta }^{\alpha }\\ {\delta }^{k_1+\alpha -3}-{\delta }^{k_2+\alpha -2}+\alpha {\delta }^{\alpha -1}=\alpha {\delta }^{\alpha -1}\end{array}\right.$$

(13)

Then, the relationship for k₁, k₂ and k₃ is derived, $k_2=k_1-1,k_3={\displaystyle \frac{k_1{ }^2-k_1+1}{k_1(k_1-1)}}-\alpha$, which is brought into the Ifal function to obtain the new nonlinear function:

$$Ifal(e,\alpha ,\delta )=\left\{\begin{array}{l}{\left|e\right|}^{\alpha }sign(e)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em},\left|e\right|>\delta \\ {\displaystyle \frac{{\delta }^{\alpha -3}{e}^{k_1}\cos (e-\delta )}{k_1}}-{\displaystyle \frac{{\delta }^{\alpha -2}{e}^{k_1-1}\cos (e-\delta )}{k_1-1}}+{\displaystyle \frac{{\delta }^{k_1+\alpha -3}}{k_1(k_1-1)}}+\left(1-\alpha \right){\delta }^{\alpha }+\alpha {\delta }^{\alpha -1}e,\left|e\right|\le \delta \end{array}\right.$$

(14)

The fal function curves before and after improvement are drawn by MATLAB 2020a from MathWorks and are shown in Figure 3.

It can be seen from the graph that the Ifal(e, α, δ) function curve is smoother than that of the fal function. The Ifal function is continuous and derivable at the points that |e| = δ. In the range of (−0.06, 0.06), the output of the Ifal function is higher than that of fal function, that is, the Ifal function converges faster and more smoothly.

3.2. Improved ESO Design (IESO)

Moreover, the IESO serves as the nucleus for IADRC. It not only provides the state estimation of the PMSM system, but also classifies all types of both internal and external disturbances, in addition to uncertainties comprising the entirety of the PMSM system’s disturbances, and estimates and compensates for the total disturbance in real time. Similar to the ESO, the IESO relies solely on the input and output data of the system and does not require an accurate mathematical model of disturbances. This approach effectively enhances the performance and robustness of the control system. The formulation of the IESO is as follows:

$$\left\{\begin{array}{l}{e}_2=z_{21}-n\\ {\stackrel{.}{z}}_{{\hspace{0.17em}}_{21}}=bu(t)+z_{22}-k_{31}Ifal(e_2,{\alpha }_2,{\delta }_2)\\ {\stackrel{.}{z}}_{{\hspace{0.17em}}_{22}}=-k_{32}Ifal(e_2,{\alpha }_2,{\delta }_2)\end{array}\right.$$

(15)

3.3. Improved NLSEF Design (INLSEF)

The NLSEF realizes the effective control of nonlinearity, uncertainty and external disturbance in ADRC and enhances the system’s control performance and robustness. The fal function is a key nonlinear element in the NLSEF for processing error signals. It can adaptively manipulate the amplification based on the magnitude of error, diminishing the gain in instances of substantial error and elevating the gain when the error is minimal, thereby enhancing the control mechanism’s accuracy and robustness. The expression of the INLSEF is as follows:

$$\left\{\begin{array}{l}{e}_1=z_{11}-z_{21}\\ u(t)=-{\displaystyle \frac{z_{22}}{b}}+k_{2}Ifal(e_1,{\alpha }_1,{\delta }_1)\end{array}\right.$$

(16)

where z₂₁ represents an observation of the controller’s dynamic response, while z₂₁ denotes an observation of the disturbance, b is the compensation factor and k₂ is the regulator gain.

3.4. Design of PMSM Speed Control System Based on IADRC-Proposed

Figure 4 depicts the schematic diagram of the IADRC-proposed PMSM speed control system. The controller indirectly controls the speed and torque of the PMSM by controlling the inverter’s output. The system adopts double closed-loop control, wherein the current loop operates under PI, while the speed loop is controlled by IADRC-proposed.

4. Simulation and Experimental Evaluation

To verify the effect of the IADRC-proposed algorithm on PMSM speed control, the traditional ADRC, SMC, fuzzy PID and the IADRC in Ref. [34] are adopted as the comparison algorithms. The PMSM speed regulation system models based on these five control algorithms are built in MATLAB/Simulink, and the simulation comparison experiments are carried out thereafter. The specifications for the PMSM in the simulation models are outlined in

Table 1. Specifications for PMSM.

Specifications	Value
Stator phase resistance R (Ohm)	2.785
Stator direct-axis inductance LD (H)	8.5 × 10−3
Stator cross-axis inductance LQ (H)	8.5 × 10−3
Permanent magnet flux linkage ${\psi }_f$ (Wb)	0.175
Inertia J (kg∙m2/rad)	1 × 10−3
Pole pairs, P	4
Coefficient of friction B	1 × 10−4

.

The methodology incorporated within this manuscript is delineated as follows: the expected speed of the PMSM is 1500 r/min. To verify the robustness of the IADRC-proposed algorithm, a load torque is added first. When the actual speed reaches the expected speed and is stable, the load torque of 2 N·m is suddenly applied to the PMSM’s output shaft at 0.1 s to compare the no-load response curve and anti-interference ability of the PMSM under the action of the above five control algorithms. The total experimental time is 0.15 s.

The parameter settings in the IADRC-proposed algorithm are as follows. According to the specific descriptions of the ADRC’s three components in Section 3, the two parameters $\alpha ,\delta$ in the TD and NLSEF are set based on experience, α₁ = 0.4, δ₁ = 0.09, α₂ = 0.4, δ₂ = 0.09. The two parameters in the ESO are determined through continuous simulation and tuning: α₃ = 0.25, δ₃ = 0.04. Velocity coefficient k₁ in the TD, regulator gain k₂ and compensation factor b of the NLSEF, derived from the mathematical model and parameters of the PMSM system, are taken as k₁ = 5355, k₂ = 5000, b = 1030.66, respectively. Similarly, the calibration gains of the output in the ESO are taken as k₃₁ = 8500, k₃₂ = 500,000. The key parameters in the controller are shown in

Table 2. The values of key parameters of controller.

Component	Specifications	Value
TD	Tracking factor ${\alpha }_1$	0.4
	Filter factor ${\delta }_1$	0.09
	Velocity factor k1	5355
NLSEF	Tracking factor ${\alpha }_2$	0.4
	Filter factor ${\delta }_2$	0.09
	Regulator gain k2	5000
	Compensation factor b	1030.66
ESO	Tracking factor ${\alpha }_3$	0.25
	Filter factor ${\delta }_3$	0.04
	The calibration gains of the output k31	8500
	The calibration gains of the output k32	500,000

.

According to the above parameter settings, the three algorithms are applied to the PMSM’s speed control. The results of the experiment are shown below:

It can be seen from Figure 5 and Figure 6 that in the PMSM’s initial start-up stage, the IADRC proposed in this paper has the best effect among the three algorithms. It not only has no overshoot but also reaches and stabilizes at the expected speed at 0.027 s. The maximum error fluctuation value is 31.28 r/min, which is about 2.09% of the expected speed. The IADRC in Ref. [34] needs 0.045 s to reach and stabilize at the expected speed, and the overshoot is 61.64 r/min, which is about 4.11% of the expected speed. And the time for the traditional ADRC to reach and stabilize at the expected speed is 0.085 s, and its overshoot is also the largest, which is 119.04 r/min, about 7.89% of the expected speed. The SMC requires 0.075 s to reach and stabilize at the expected speed, with a maximum overshoot of 423.71 r/min, which is approximately 28.25% of the expected speed. The fuzzy PID takes 0.043 s to reach and stabilize at the expected speed, with a maximum error fluctuation of 30.27 r/min, which is about 2.02% of the expected speed.

When the speed of the PMSM is stabilized at the expected speed, the step load torque is applied at 0.1 s. The IADRC algorithm proposed in this paper is basically unaffected, and the speed curve only fluctuates slightly. The maximum speed error is 7.39 r/min, which is about 0.49% of the expected speed, and then quickly (0.004 s) returns to the expected value. The speed curve of the IADRC algorithm in Ref. [34] produced a maximum speed error of 20.73 r/min, which is about 1.38% of the expected speed, and returned to the expected value after 0.006 s. And the speed curve of the traditional ADRC algorithm produces a maximum speed error of 29.85 r/min, which is about 1.99% of the expected speed, and returns to the expected value after 0.034 s. The velocity curve of the SMC algorithm produces a maximum speed error of 81.98 r/min, which is approximately 5.47% of the expected speed, and returns to the expected value after 0.028 s. The velocity curve of the fuzzy PID algorithm produces a maximum speed error of 18.11 r/min, which is about 1.21% of the expected speed, and returns to the expected value after 0.07 s.

Various uncertain factors affect the PMSM system in practical work, such as environmental noise and resistance changes caused by continuous operation of the motor or changes in ambient temperature. To verify the robustness of IADRC-proposed algorithm under different parameter variations and uncertainties, simulation experiments for noise interference and motor resistance change are added to the system.

To verify the robustness of the IADRC-proposed algorithm, this paper also includes comparative experiments of the plant control input U under Gaussian noise, comparing it with other algorithms. The Gaussian noise is added at the beginning, and the signal is set to have a mean of 1 and a variance of 200. Other motor parameters are shown in Table 1 and Table 2. The simulation results are shown in Figure 7 and Figure 8.

Comparing Figure 5, Figure 6, Figure 7 and Figure 8, it is obvious that the speed response curves and the system output U do not change significantly before and after the Gaussian noise is added for all five algorithms. Moreover, the speed response curves and the system output U controlled by the IADRC-proposed algorithm have the shortest oscillation time and the smallest oscillation amplitude, which verifies the stability of the IADRC-proposed algorithm.

To further validate the impact of motor resistance changes on the IADRC-proposed, the motor resistances are set to 2.5875, 2.875 and 3.1625, respectively. Other parameters are consistent with Table 1. The experimental results are shown in Figure 9.

From Figure 9, it is obvious that although the overshoot and the time to reach stability all change, the change is slight. It indicates that after changes in external conditions, the control effect is still very good, confirming the robustness of the IADRC-proposed algorithm.

In summary, it is effectively proved that our presented IADRC exhibits enhanced responsiveness, higher control accuracy, stronger robustness and anti-interference ability. The specific performance is as follows: our proprietary IADRC algorithm is 40.00% better than the IADRC in Ref. [34], 68.24% better than the traditional ADRC, 64.00% better than SMC and 37.21% better than fuzzy PID in terms of rapidity. In terms of accuracy, it is 49.25% better than the IADRC in Ref. [34], 73.72% better than the traditional ADRC, 92.62% better than SMC and 3.35% worse than fuzzy PID. Moreover, when the step load torque is applied in the stable operation state, the IADRC algorithm proposed in this paper is 33.33% better than the IADRC in Ref. [34], 88.24% better than the traditional ADRC, 85.71% better than SMC and 94.29% better than fuzzy PID in terms of rapidity. In terms of accuracy, it is 64.35% better than the IADRC in Ref. [34], 75.24% better than the traditional ADRC, 90.99% better than SMC and 59.19% better than fuzzy PID. Additionally, experimental tests adding Gaussian noise and changing motor resistance attest that the proposed IADRC algorithm has good robustness.

5. Conclusions

A refined fal function-based IADRC strategy is elucidated in this article, and a better control performance for the PMSM is obtained compared to the traditional ADRC, IADRC in Ref. [34], SMC and fuzzy PID. Simulation results indicate that compared to ADRC, IADRC in Ref. [34], SMC and fuzzy PID, the IADRC proposed in this paper has faster response speed. The time to reach and stabilize at the expected speed is 68.24% shorter than that of ADRC, 40.00% shorter than that of IADRC in Ref. [34], 64.00% shorter than that of SMC and 37.21% shorter than that of fuzzy PID. Moreover, the IADRC proposed in this paper has no overshoot, and the maximum error reduction amounts to 73.72% compared to ADRC, 49.25% compared to IADRC in Ref. [34], 92.62% compared to SMC and −3.35% compared to fuzzy PID. In the case of sudden disturbance when the speed of the PMSM is stabilized at the expected speed, the time to reach and stabilize at the expected speed is 88.24% shorter than that of ADRC, 33.33% shorter than that of IADRC in Ref. [34], 85.71% shorter than that of SMC and 94.29% shorter than that of fuzzy PID. Additionally, the maximum error of the IADRC proposed in this paper is 75.24% lower than that of ADRC, 64.35% lower than that of IADRC in Ref. [34], 90.99% lower than that of SMC and 59.19% lower than that of fuzzy PID. Apart from that, the good robustness is also verified by adding Gaussian noise and changing motor resistance.

Our future work will focus on applying the IADRC-proposed algorithm to the physical control of PMSMs and continue to improve the algorithm according to the actual control situation.