Frontiers in Computing and Intelligent Systems ISSN: 2832-6024 | Vol. 11, No. 1, 2025 12 Research on Sensorless FOC Method of PMSM based on ADRC Xin Zhang 1, 2, Miao Huang 1, *, Lili Tao 1, Shuping Zhang 1, Zhihua Hu 1 1 School of Intelligent Manufacturing and Control Engineering, Shanghai Polytechnic University, Shanghai, 201209, China 2 Vocational & Technical Teacher Education School, Shanghai Polytechnic University, Shanghai, 201209, China * Corresponding author: Miao Huang (Email: huangmiao@sspu.edu.cn) Abstract: This paper addresses the issues of low observer estimation accuracy, high speed overshoot, narrow speed range, and poor anti-disturbance performance in traditional sensorless Permanent Magnetic Synchronous Motor (PMSM) Field Oriented Control (FOC) systems. It proposes two control methods combining the Extended Kalman Filter (EKF) observer with Linear Active Disturbance Rejection Control (LADRC) and Nonlinear Active Disturbance Rejection Control (NLADRC). The EKF is used to estimate rotor position and speed, while LADRC and NLADRC controllers compensate for load disturbances in the speed loop. EKF outputs are used for feedforward compensation in the current loop, eliminating the coupling between d-axis and q-axis voltages. Simulation results show that both methods offer strong anti-disturbance performance, a wide speed range, strong speed overshoot suppression, and accurate estimation of motor speed and rotor position, thereby enhancing the operational stability of the PMSM system. Keywords: PMSM; EKF; ADRC; FOC; Voltage Feed-forward Control; Sensorless Control. 1. Introduction PMSM (Permanent Magnet Synchronous Motors) are widely used in industrial control, medical machinery, automotive manufacturing, and other fields due to their advantages such as simple structure, high power density, and low manufacturing cost. Traditional PMSM control strategies include sine pulse width modulation (SPWM), Field-Oriented Control (FOC), Direct Torque Control (DTC), etc. Among these, FOC technology offers advantages like fast response, high voltage utilization, a wide speed range, and high control accuracy. However, traditional FOC relies on position sensors to obtain rotor angle, which increases motor cost and size, and significantly reduces motor adaptability. To solve these issues, sensorless FOC has emerged. Sensorless FOC utilizes state observation algorithms to establish rotor speed and position observers to replace position sensors. Common state observation algorithms used in sensorless FOC include the Extended Kalman Filter (EKF) observer, Luenberger observer, and others. The EKF observer is based on the minimum variance estimation theory and achieves accurate estimation of state variables through data fusion. It can operate over a wide speed range, has filtering capabilities, and is adaptive with low requirements for initial state estimation. For instance, Changchang Jiang et al. proposed a sensorless vector control strategy based on the EKF observer and used a Luenberger observer to track load changes, providing feedforward compensation for the current loop PI controller, thereby enhancing the tracking and disturbance rejection capabilities of the EKF observer [1]. Shijiao Wang et al. proposed a particle swarm optimization-based fuzzy PI control method, which improves the dynamic and disturbance rejection performance of the system[2]. Weilin Yang et al. proposed a control strategy for PMSM based on an improved neural network[3]. Lin Liu et al. proposed a PI control parameter self-tuning and optimization method combining model and rule-based approaches[4]. Jiaxuan Li et al. designed an adaptive PI controller based on BP neural networks and improved the traditional FOC model to enhance system response speed and robustness[5]. These scholars have used modern control theory and intelligent algorithms to improve traditional PI controllers. Although the improvements are significant, the contradiction between overshoot and speed remains unresolved. Furthermore, the rapid development of modern control theory has led to the design of various motor control algorithms, such as sliding mode control, model reference adaptive control, robust control, internal model control, and disturbance-free control, to replace traditional PI control [6,7,8]. In the 1980s, Professor Jingqing Han analyzed the drawbacks of modern control theory's over-reliance on mathematical models and studied traditional control methods for error suppression through system feedback[9]. In 1998, he proposed the Active Disturbance Rejection Control (ADRC) theory and published a monograph[10]. This control method not only has a simple algorithm but also detects and compensates for internal and external disturbances, enabling the system to promptly respond to disturbances and parameter changes. ADRC is characterized by strong disturbance suppression, high robustness, and good operability[11,12]. For example, Shiwei Chi et al. proposed a feedforward compensation-based ADRC strategy for PMSMs to suppress speed fluctuations caused by sudden load torque changes, optimizing the control performance[13]. Zixin Li et al. proposed a parameter self-tuning strategy based on BP neural networks to address the difficulty of tuning ADRC parameters, effectively reducing the complexity of parameter tuning while improving system performance and lowering the difficulty of applying ADRC in engineering[14]. Yicheng Wang et al. proposed an improved PMSM weak magnetic ADRC method to meet the high-speed, high-stability, and fast-response requirements of multi-electric aircraft, and demonstrated its superior performance through comparisons with traditional weak magnetic control and ADRC weak magnetic control [15]. This paper proposes a sensorless ADRC strategy for 13 PMSMs based on the EKF observer to address issues such as load disturbance impact on PMSM system stability, the trade- off between response speed and overshoot in PMSM systems, and the high cost and poor adaptability caused by position sensors. The speed loop replaces the PI controller with an ADRC controller, and the EKF observer estimates speed and angle. The current loop implements decoupling of the d-axis and q-axis voltages through feedforward compensation. Finally, comparative experiments are conducted on three control strategies: PI + FOC + EKF, LADRC + FOC + EKF, and NLADRC + FOC + EKF, to verify the control effect of the proposed method. 2. Mathematical Model of PMSM The following assumptions are made for the PMSM model: (1) The rotor permanent magnet's magnetic field distribution is sinusoidal. (2) The back electromotive force (EMF) in the stator windings is sinusoidal. (3) The stator core saturation, as well as eddy current and hysteresis losses in the core, are neglected. (4) There are no damper windings on the rotor. The stator voltage equation of the PMSM in a synchronous rotating coordinate system is: d d d e q q q q e d d u Ri dt d u Ri dt                (1) The flux linkage equations are as follows: d d d f q q q L i L i        (2) and are the voltage components of the stator in the axis and axis , respectively; and are the current components of the stator in the axis and axis , respectively; and are the flux linkage components of the stator in the axis and axis , respectively; and are the inductance components of the stator in the axis and axis , respectively; , and are the stator resistance, electrical angular velocity, and permanent magnet flux linkage, respectively. The electromagnetic torque equation of the motor is: ( ) [ ( ) ]e n d q q d n f q d q d qT P i i P i L L i i       (3) is the pole number, and is the electromagnetic torque. The conversion relationships between the mechanical angular velocity (rad·s⁻¹), mechanical speed (r·min⁻¹), electrical angular velocity (rad·s⁻¹), and electrical angle (rad) are as follows: 30 e n m r m e e P N dt                (4) is the mechanical angular velocity of the motor, is the mechanical speed of the motor, and is the electrical angle of the motor. The mechanical motion equation of the motor is: m e L m d J T T B dt     (5) is the moment of inertia; is the damping coefficient; is the load torque. This paper combines the EKF algorithm and ADRC theory to propose a sensorless control method for PMSM. The rotor position and speed are estimated through EMF, enhancing the system's stability and accuracy, while addressing the trade-off between response speed and overshoot. 3. The Design of the Improved FOC Control Scheme The core of FOC is to use coordinate transformation to degrade and decouple the complex and strongly coupled PMSM model in the natural coordinate system. Ultimately, the motor model is transformed into the PMSM model shown in equations (1) to (5), achieving decoupled control of the excitation current and torque current. The main steps are as follows: (1) The equal-magnitude Clarke transform converts the sampled three-phase current into a two-phase stationary coordinate system, represented as , . The formula for the equal-magnitude Clarke transform is as follows: 2 ( cos0 cos120 cos240 ) 3 2 ( sin 0 sin120 sin 240 ) 3 a b c a b c i i i i i i i i                         (6) After simplification, the Clarke transform formula becomes: 2 1 1 ( ) 3 2 2 2 3 3 ( ) 3 2 2 a b c b c i i i i i i i           (7) (2) Through the Park transform, using the rotor angle information , obtained by the observer or sensor, the current components , in the coordinate system are transformed into the coordinate system, which is aligned with the rotor flux position, resulting in and . The Park transform formula is as follows: cos sin sin cos d q i i i i i i                   (8) (3) The differences between , and the current reference values , from the speed loop output are sent into the current loop controller to calculate the next voltage vectors and . aligns with the rotor flux to control the flux, and is generally set to 0. is perpendicular to the rotor flux to control the output torque. (4) Using the inverse Park transformation, the newly estimated electrical angle is used to transform and into the two-phase stationary reference frame, resulting in and . The formula for the inverse Park transformation is as follows: cos sin sin cos d q d q V V V V V V                (9) (5) Through the Clarke inverse transformation, and are converted into , and in the natural coordinate system, and then sent to the SVPWM module to generate the SVPWM wave, which is output to the three-phase inverter bridge. The formula for the Clarke inverse transformation is as follows: 14 sin30 cos30 sin30 cos30 a b c V V V V V V V V                        (10) After simplification, it becomes: 1 3 2 2 1 3 2 2 a b c V V V V V V V V                    (11) (6) In the speed loop, the speed information obtained by the observer and the error between the given speed reference are fed into the speed loop controller to obtain . From Equation (3), it can be seen that the motor model after the FOC transformation is not completely decoupled, as and contain components of each other. This results in the inability to achieve independent control and causes high- frequency oscillations. Therefore, by introducing feedforward compensation signals equal to the coupling terms into the outputs of the axis and axis controllers, decoupled control of torque and magnetizing current is achieved. The control system block diagram is shown in Figure 1: Figure 1. Block diagram of the improved FOC control system 4. Design of Speed and Position Observer Based on EKF Algorithm The PMSM system is a typical nonlinear system, and its state-space expression is: [ ] [ 1] [ 1] [ 1]( , , )k k k kx f x u w   (12) [ ] [ ] [ ]( , )k k kz h x v (13) In this expression, and are nonlinear equations, ~ 0, and ~ 0, represent the system noise and measurement noise, respectively, which follow a normal distribution. 4.1. The Core Algorithm of the EKF Observer The EKF observation algorithm is divided into two main steps: prediction and correction: Prediction stage (time update stage): (1) Calculate the prior state estimate. [ ] [ 1] [ 1]ˆ ˆ( , ,0)k k kx f x u   (14) is the prior state estimate at time k, is the posterior state estimate at time k-1, and is the system input at time k-1. In the prior state estimation, the system error is assumed to be zero. The initial state estimate is denoted as . (2) Calculate the Jacobian matrix (linearization of the nonlinear system). The EKF algorithm uses the Taylor series expansion method to linearize the nonlinear system. The equation (12) is expanded in a Taylor series around the posterior state estimate , ignoring the higher-order terms in the Taylor expansion and assuming that the system error around is zero. [ ] [ 1] [ 1] [ 1] [ 1] [ 1] [ ] [ 1] [ 1] [ ] [ 1] ( , , ) ˆ ˆ( , ,0) ( ) ( 0) k k k k k k k k k k k x f x u w f x u A x x W w               (15) [ ] [ 1] [ 1]ˆ( , ,0)k k k f A x u x      (16) [ ] [ 1] [ 1]ˆ( , ,0)k k k f W x u w      (17) Equations (16) and (17) represent the Jacobian matrices of the first-order partial derivatives of with respect to the system state and the system error , respectively. (3) Calculate the covariance matrix of the prior state estimate error. [ ] [ ] [ ] [ 1] [ ] [ ] T T k k k k k kP A P A W QW   (18) The covariance matrix of the system noise and the covariance matrix of the initial state estimation error are denoted as and . Correction stage (measurement update): (1) Calculate the Jacobian matrix. Expand equation (13) around , that is, [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]ˆ ˆ( , ) ( ,0) ( ) ( 0)k k k k m k k k k kz h x v h x H x x V v       (19) [ ] [ ]ˆ( ,0) km k h H x x    (20) [ ] [ ]ˆ( ,0)k k h V x v    (21) and are the Jacobian matrices of with respect to and , respectively. Equations (15) and (19) form the system state-space equations after linearization, which can be simplified as: [ ] [ ] [ 1] [ ] [ 1] [ ] [ ] [ 1]ˆ ˆ( )k k k k k k k kx A x W w x A x       (22) [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ]ˆ ˆ( ( ,0) ) k kk m k k k k m kz H x V v h x H x     (23) This is a linear equation, so the noise 15 [ ] [ 1] [ ] [ ]~ (0, )T k k k kW w N W QW , [ ] [ ] [ ] [ ]~ (0, )Tk k k kV v N V RV are assumed to follow a normal distribution. (2) Calculate the Kalman gain. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] k k k T k m k T T m k m k k P H K H P H V RV    (24) (3) Calculate the posterior state estimate. [ ] [ ] [ ] [ ] [ ]ˆ ˆ ˆ[ ( ,0)]k k k k kx x K z h x    (25) (4) Update the posterior state estimate error covariance matrix. [ ][ ] [ ] [ ]( ) kk k m kP I K H P  (26) Equations (14) and (25) use the noiseless nonlinear system model to obtain the prior and posterior estimates. Equations (18), (24), and (25) all use the linearized model for calculations. 4.2. Design of the Speed and Position Observer The EKF is a state observer that relies on the motor model and can perform estimations in both synchronous rotating and stationary reference frames. The synchronous reference frame requires multiple coordinate transformations and involves trigonometric functions (sine and cosine), which increase nonlinearity and computational complexity, thereby extending the recursion time. In contrast, the stationary reference frame only requires a Clarke transformation, which reduces the model's nonlinearity and improves estimation accuracy. The structure diagram of the EKF observer is shown in Figure 2. Figure 2. Block Diagram of Extended Kalman Filter Observer Structure The voltage equations of the PMSM in the stationary reference frame is: sins e f e di u Ri L dt         (27) coss e f e di u Ri L dt         (28) The relationship between the electrical angle and the electrical angular velocity is as follows: 0e e e d dt d dt          (29) Equations (27), (28), and (29) represent the state equations of the PMSM system in the stationary reference frame. Let , , , represent the state variables of the observation system, , the input of the observation system, and , the output of the observation system. The state equations of the PMSM can then be written as: ( )x f x Bu    (30) In the equation: represents the system noise. 0 0 0 0 0 0 represents the input matrix, sin cos 0 . Since and can be directly measured, the measurement equation is defined as: y Cx v  (31) In the equation: 1 0 0 1 0 0 0 0 represents the output matrix. The Jacobian matrix of the system's state equations is: ( ) ( ) ( ) ˆ ( ) ( ) ˆ ˆˆ0 sin cos ( ) ˆ ˆ0 ˆ( ) | cos sin 0 0 0 0 0 0 1 0 k f f e k e e k s s s f f x x e k e e k s s s R L L L Rf x F k Lx L L                                  4.3. The Selection of the Noise Covariance Matrix In the EKF algorithm's estimation process, the covariance matrices of the noise and are required. The definitions of the covariance matrices and are as follows:     T T Q E WW R E VV     (32) The initial state random vector and the noise vectors and are mutually independent. Since the off-diagonal elements have a negligible impact on the system, , and are generally taken as diagonal matrices. From equations (18) and (24), it is known that the matrix and the matrix adjust the Kalman gain, affecting the weight of the estimated values and the measurement values. Increasing leads to a higher Kalman gain, increasing the weight of the measurement values; increasing leads to a lower Kalman gain, increasing the weight of the estimated values. A larger allows the system to converge faster but becomes less reliable; a larger causes the observer's output to lag, deteriorating dynamic performance. The PMSM model is complex, and selecting the appropriate and matrices is difficult. Particle swarm optimization (PSO), grey wolf optimization (GWO), and other algorithms can be used to optimize the elements of the and matrices. In simulations, the EKF outputs the optimal estimated values of the motor model, not the actual optimal estimates of the real motor system, so there may be some deviation between the estimated values and the actual feedback values. 5. Design of the Speed Loop Controller Based on ADRC The tracking differentiator (TD), extended state observer 16 (ESO), and nonlinear state error feedback control law (NLSEF) form the active disturbance rejection controller (ADRC) [16]. Its structure is shown in Figure 3. Figure 3. NLADRC controller structure diagram The TD quickly tracks the desired signal using the fastest synthesis function and synchronizes the output differential signal, reducing noise amplification and steady-state errors. By performing multi-order planning on the desired signal, it reduces the impact of the input signal on the system, achieving smooth input and thereby reducing overshoot. The discrete mathematical model of the TD is: 1 2 0 0 1 1 2 2 2 ( ( ) ( ), ( ), , ) ( 1) ( ) ( ) ( 1) ( ) fh fhan x k v k x k r h x k x k hx k x k x k hfh           (33) Where: 2 0 0 0 0 2 1 0 1 1 2 0 1 2 0 0 0 2 2 0 ( 8 ) ( )( ) 2 ( , , , ) ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) y y a a d r h a h x y x a a d d y sign y a d a a fhan x x r h sign y d sign y d S a a y a S a sign a d sign a d S a fhan r sign a S r sign a d                                           (34) , are the target control variables and their derivatives, , are the control parameters of the function. controls the tracking speed: the larger is, the faster the tracking, but the noise amplification effect and overshoot will also increase. If is too large, it may cause system oscillation. Typically, in the PMSM system, ∈ 0,1000 is acceptable. controls the internal iteration step size of , which is similar to the system sampling period. To avoid noise amplification, is usually selected as an integer multiple of the sampling period. The discrete mathematical model of the nonlinear Extended State Observer (ESO) is: 1 1 1 2 01 2 2 3 02 1 3 3 03 2 ( ) ( ) ( ) ( 1) ( ) [ ( ) ( )] ( 1) ( ) [ ( ) ( , , ) ] ( 1) ( ) ( , , ) e k z k y k z k z k h z k e k z k z k h z k fal e a bu z k z k h fal e a                       (35) Where, is the nonlinear function: (1 ) , ( , , ) ( ) , a a x x fal x a sign x x x           (36) The function , , ensures that the ESO state variables can closely approximate the system state variables , and have a larger adaptation range. The parameters that need to be tuned in equation (35) include: , , , , , , . and are nonlinear factors, with values in the range ∈ 0,1 , and . is the filtering factor, with values in the range ∈ 0,0.1 . The error gain parameters , , need to be analyzed based on the specific system characteristics. Generally, the relationship among their values is: 02 01 2 03 01 10       (37) The compensation factor needs to be analyzed based on the output waveform of the controller. Increasing can reduce chattering caused by an increased controller bandwidth, but an excessively large will decrease the disturbance compensation amount, reducing the effectiveness of disturbance suppression. The combination relationship of the NLSEF is: 0 1 1 1 2 2 2( , , ) ( , , )u fal e a fal e a     (38) The tunable parameters in NLSEF include the error gains and . Increasing and accelerates the speed response, but excessive values can lead to overshoot, while smaller values may result in steady-state errors in speed. During parameter tuning, increments are usually made in orders of magnitude until overshoot appears, and then the values are gradually reduced to the optimal level. The tuning methods for , and are similar to those used in ESO, but in NLSEF, the value range of is ∈ 0,1 , and for , ∈ 1, ∞ . Linear Active Disturbance Rejection Control (LADRC) simplifies the NLADRC by replacing the TD and NLSEF with PD control. This approach associates the tunable parameters with the frequency-domain bandwidth of the controller, simplifying the controller design process. The basic structure of LADRC is shown in Figure 4. Figure 4. LADRC controller structure diagram The parameter tuning of NLADRC and LADRC controllers is relatively challenging. Optimization algorithms such as Whale Optimization Algorithm (WOA), Grey Wolf Optimization (GWO), and Reinforcement Learning (RL) can be used for automatic parameter tuning, significantly reducing the workload involved in the tuning process. 17 6. Simulation Testing and Results Analysis 6.1. Construction of the Simulink Simulation Model Based on the above content, construct a graphical simulation control model for PMSM in the MATLAB/Simulink environment. Perform comparative simulation experiments for three control systems: PI+EKF, LADRC+EKF, and NLADRC+EKF. The model uses a fixed step size of 2e-6s, the ode23tb solver, and a simulation duration of 0.6s. Tables 1~4 detail the parameters of each part of the model, and the model structure is illustrated in Figures 5~7. Figure 5. Simulation model diagram of PI+EKF control system Figure 6. Simulation model diagram of LADRC+EKF control system Figure 7. Simulation model diagram of NLADRC+EKF control system 18 Table 1. Motor Simulation Model Parameter Table Name Value DC-side supply voltage (Vdc/V) 311 Number of motor pole pairs (Pn) 4 Per-phase stator resistance (R/Ω) 2.8750 Moment of inertia J/(kg·m-2) 0.001 Stator inductance L/mH 0.0085 Rotor flux linkage ψ/Wb 0.175 Table 2. ESO parameter table Name Value Q11 1.5 Q22 1.5 Q33 26 Q44 0.2 P11 0.1 P22 0.1 R11 20.9 R22 20.9 Table 3. LADRC parameter table Name Value b0 1096 Kp 0.17 3600 3239049 Table 4. NLADRC parameter table Name Value R0 600 h0 0.01 a1 0.5 a2 0.25 0.015 300 3520 115300 300 11000 60 6.2. Simulation Results The initial speed for all three models is set to 1500 RPM. At 0.2s, the load increases by 2 N·m, and at 0.4s, the speed is reduced to 1000 RPM. The speed waveform comparison is shown in Figure 8. Figure 8. Comparison of motor speed waveforms for the three control strategies As shown in the figure, when the reference speed is set to 1500 RPM within the first 0.4 seconds: PI+EKF system overshoots to 1821 RPM at 0.0089 seconds and stabilizes at 1500 RPM in 0.1 seconds; LADRC+EKF system exhibits no overshoot and stabilizes at 0.047 seconds; NLADRC+EKF system also shows no overshoot and stabilizes at 0.035 seconds. At 0.2 seconds, when a 2 N·m load is applied: PI+EKF system takes 0.07 seconds to return to 1500 RPM; LADRC+EKF system takes 0.016 seconds; NLADRC+EKF system takes only 0.002 seconds. At 0.4 seconds, when the reference speed drops to 1000 RPM: PI+EKF system responds the fastest but overshoots to 906 RPM and stabilizes in 0.053 seconds; LADRC+EKF and NLADRC+EKF systems show no overshoot and stabilize in 0.039 seconds and 0.02 seconds, respectively. Figures 9 to 11 illustrate the current waveforms of the PI+EKF, LADRC+EKF, and NLADRC+EKF systems, respectively. From Figures 9 to 11, under a load of 3 N·m during motor start-up: PI+EKF system's current stabilizes after 0.052 seconds, with significant overshoot; LADRC+EKF system exhibits reduced current overshoot and stabilizes after 0.035 seconds; NLADRC+EKF system shows the smallest current overshoot and stabilizes after 0.033 seconds. When the reference speed drops to 1000 RPM at 0.4 seconds: PI+EKF system's current stabilizes in 0.024 seconds; LADRC+EKF system stabilizes in 0.02 seconds; NLADRC+EKF system stabilizes in 0.022 seconds. In summary, the NLADRC+EKF system demonstrates superior performance but poses greater challenges in parameter tuning. The LADRC+EKF system performs 19 slightly less effectively but is easier to tune, making it more convenient for engineering applications. Figure 9. current waveform based on PI+EKF system Figure 10. current waveform based on LADRC+EKF system Figure 11. current waveform based on NLADRC+EKF system Figure 12. Comparison of EKF estimated angular velocity and the real angular velocity 20 Figure 13. Comparison of the EKF estimated rotor electrical angle with the true electrical angle Figures 12 and 13 respectively show the comparison between the EKF-estimated and actual angular velocity, as well as the comparison between the actual and estimated electrical angles. Figure 12 shows that during the first 0.049 seconds, the estimated speed value lags behind the actual value, and after that, the error approaches zero. Figure 13 shows that the estimated rotor electrical angle fits well with the actual electrical angle from the start, indicating that the EKF is able to track the rotor position effectively. Due to the excellent dynamic stability and compensation performance of the ADRC controller, the EKF estimation will become increasingly closer to the actual value as the operating time increases. 7. Experimental Verification 7.1. Experimental Verification To verify the performance of the controller designed in this paper, an experimental setup was built as shown in Figure 14. The main control chip used is the STM32F407IGT6. The inverter adopts a three-phase bridge inverter circuit. The control board is connected to a PC host via a serial port, allowing real-time transmission of motor parameters to the host for graphical display. The controller is designed and the parameters are tuned as described in the previous sections. Figure 14. Diagram of the experimental platform The PMSM motor parameters used in the experiment are shown in Table 5. Table 5. PMSM Parameter Table Parameter Value Rated power Pe/W 100 Rated voltage Ve/V 24 Rated speed n/RPM 4500 Rated current Ie/A 6 Stator resistance Rs/Ω 0.42 Number of pole pairs Pn 4 Stator inductance Ls/mH 1.24 Rated torque (N·m) 0.38 7.2. Experimental Results The target speed of the motor is set to 4500 RPM. At 3 seconds, the load is increased by 2 N·m. From 6 seconds to 20 seconds, the target speed of the motor gradually decreases from 4500 RPM to 500 RPM in steps of 1000 RPM, and then stops. The speed waveforms for the LADRC+EKF and NLADRC+EKF systems are shown in Figures 15 and 16, respectively. Figure 15. Speed waveform based on LADRC+EKF system As shown in Figure 15, within the first 6 seconds, when the reference speed is set to 4500 RPM, the LADRC+EKF system exhibits no overshoot, and the speed stabilizes at 4500 RPM in 0.7 seconds. At 3 seconds, a 2 N·m load is applied. The LADRC+EKF system takes 1.2 seconds to return to 4500 RPM, with an overshoot of 4560 RPM during the recovery process. From 6 seconds onwards, the target speed gradually decreases by 1000 RPM steps down to 500 RPM. The LADRC+EKF system follows each speed change in 0.2 seconds. However, when the speed decreases to 1500 RPM and 500 RPM, there is a noticeable overshoot. Subsequently, when the motor stops, the speed change is smooth, and no 21 reversal occurs. Figure 16. Speed waveform based on NLADRC+EKF system As shown in Figure 16, within the first 6 seconds, when the reference speed is set to 4500 RPM, the NLADRC+EKF system exhibits no overshoot, and the speed stabilizes at 4500 RPM in 0.9 seconds. At 4 seconds, 2 N·m load is applied. The NLADRC+EKF system takes 0.6 seconds to return to 4500 RPM without any overshoot during the recovery process. From 7 seconds onwards, the target speed gradually decreases by 1000 RPM steps down to 500 RPM. The NLADRC+EKF system follows each speed change in about 0.3 seconds, without overshoot during the process. Subsequently, when the motor stops, the speed change is smooth, and no reversal occurs. 8. Conclusion This paper focuses on PMSM and proposes two novel sensorless control methods for PMSM by combining the EKF observation algorithm and the Active Disturbance Rejection Control (ADRC) theory. Simulation and experimental results show that both LADRC+EKF and NLADRC+EKF systems outperform the PI+EKF system in terms of disturbance rejection performance, compensation ability, speed regulation range, parameter estimation accuracy, and control precision. 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