Permanent magnet synchronous motor sensorless control method based on improved anti-interference predictive phase-locked loop

Abstract

The invention discloses a permanent magnet synchronous motor sensorless control method based on an improved anti-interference prediction phase-locked loop. The anti-interference prediction phase-locked loop comprises a phase discriminator module and an improved anti-interference prediction phase-locked loop module. The improved anti-interference prediction phase-locked loop module comprises an integration module, a gradient search module, an equivalent error cost function selection module, an iteration module and a differential module. According to the invention, a cost function based on a gradient search method is designed to eliminate the interference amount existing in an input signal, the robustness of the phase-locked loop to harmonic waves and DC bias is improved, and a dual-cost function based on a Newton iteration method is designed to reduce the calculated amount of predicting the phase-locked loop. And high-performance operation of the permanent magnet synchronous motor without position sensor control is realized.

Description

[0001] This invention belongs to the field of permanent magnet synchronous motor control, and particularly relates to an anti-interference predictive phase-locked loop, a method and system for estimating the speed and rotor position of a permanent magnet synchronous motor. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in various industrial fields due to their high efficiency, high performance, and relatively simple structure. With the development of high-performance magnetic materials, power electronics technology, and modern control theory, especially the introduction of high-performance control strategies such as vector control and direct torque control, PMSM speed control systems have developed rapidly. However, high-performance PMSM speed control relies on sensor devices and precise detection technology. Traditional control systems often use mechanical sensors such as photoelectric encoders and rotary transformers to obtain rotor position information. These mechanical sensors not only increase the complexity of the system's mechanical structure but also affect the system's dynamic and static performance, reduce its robustness and reliability, and significantly increase costs. Therefore, research on sensorless control technology for PMSMs has rapidly become a hot topic.

[0003] Currently, sensorless control strategies can be broadly categorized into two types: model-based methods and saliency-based methods. The former is primarily applied in high-speed applications, typically introducing an observer to estimate back electromotive force or flux linkage information to obtain speed and angle information. The latter is applied in low-speed applications, mainly involving high-frequency signal injection. After obtaining position information, a phase-locked loop (PLL) is used to acquire motor speed and rotor position information.

[0004] In the reverse operation of a permanent magnet synchronous motor (PMSM), the rotor position estimated by a traditional quadrature phase-locked loop (quadrature phase-locked loop) will produce a steady-state error of 180 degrees compared to the actual value, leading to control failure. Furthermore, acceleration and deceleration of the PMSM will introduce a steady-state error to the rotor position. Therefore, traditional quadrature phase-locked loops have significant drawbacks in situations requiring frequent forward and reverse rotation and frequent acceleration and deceleration of PMSMs. While traditional predictive phase-locked loops (PLLs) omit the need for tuning the parameters of a traditional fixed PI controller, offering good dynamic performance and robustness, they require 64 enumeration iterations to reach the optimal position, resulting in a significant computational burden. Moreover, compared to quadrature phase-locked loops, predictive phase-locked loops exhibit larger error ripples in their static response. Summary of the Invention

[0005] To address the problems of existing technologies, this invention provides a sensorless control method for permanent magnet synchronous motors (PMSMs) based on an improved anti-interference predictive phase-locked loop (PLL). First, the position signal output by the sensorless control technology is sampled. Then, a cost function based on gradient search is designed to suppress interference such as DC bias and high-frequency harmonics in the position signal. Finally, by optimizing the iterative cost function, the number of iterations in the PLL iteration stage is reduced from 64 in traditional methods to 3, thereby reducing the computational load of sensorless control of the PMSM. Ultimately, high-performance operation of the sensorless control of the PMSM can be achieved.

[0006] To solve the aforementioned technical problems, the present invention provides the following technical solution:

[0007] To address the problems of high computational burden and large error ripple in traditional predictive phase-locked loops (PLLs) and ensure that permanent magnet synchronous motors (PMSMs) always operate in a safe and efficient state, this invention provides a sensorless control method for PMSMs based on an improved anti-interference predictive PLL. The method comprises a phase detector module 1 and an improved anti-interference predictive PLL module 2, wherein the improved anti-interference predictive PLL module 2 includes an integration module 2-1, a gradient search module 2-2, an equivalent error cost function selection module 2-3, an iteration module 2-4, and a differentiation module 2-5.

[0008] Furthermore, the phase detector module 1 is used to identify back electromotive force information. With rotor position θ e The relationship between them is expressed as follows, and the input is the stator current (i) of the motor in a two-phase stationary coordinate system. α i β ), motor stator voltage in two-phase stationary coordinate system (u) α u β This invention employs a first-order sliding mode observer to observe the back electromotive force of a permanent magnet synchronous motor, and its output is the back electromotive force. The calculation process is as follows:

[0009]

[0010] in Estimate the current for the α-axis. To estimate the current along the β axis, L d L q These are the d-axis and q-axis inductances (Wb) of the permanent magnet synchronous motor, respectively. The electric angular velocity is an estimated value, with the symbol "·" representing the differential operator and "^" representing the estimated value. The sliding mode equivalent control function is designed, and its calculation process is as follows:

[0011]

[0012] Where z α and z β This is the control input of the SMO, where `sign` is the sign function and `k` is the gain of the first-order sliding mode observer. The back EMF signal is then filtered by the LPF. The calculation process is as follows:

[0013]

[0014] These are the estimated back electromotive forces along the α-axis and β-axis, respectively. To estimate the difference between the α-axis current and the measured value To estimate the difference between the β-axis current and the measured value k is the sliding mode gain, ω c This is the cutoff frequency of the low-pass filter.

[0015] Furthermore, the improved anti-interference prediction phase-locked loop module 2 includes an integration module 2-1, a gradient search module 2-2, an equivalent position error module 2-3, a cost function selection module 2-4, an iteration module 2-5, and a differentiation module 2-6.

[0016] Furthermore, the input to the integrator module 2-1 is the back electromotive force. The integration module is used to integrate the back electromotive force information to obtain the estimated rotor flux linkage value. Its output is the input of gradient search module 2-2. The calculation process is as follows:

[0017]

[0018] in, This indicates the output estimate of the rotor flux linkage, ψ. f For permanent magnet flux linkage, A α A β Estimated back electromotive force along the α and β axes respectively The lumped interference, θ e Indicates the actual rotor position.

[0019] Furthermore, the input to gradient search module 2-2 is the output of integration module 2-1. The output quantity (q) α q β Select the input quantities for module 2-3 for the cost function. The calculation process is as follows:

[0020]

[0021] Where μ is the gain of the gradient search, The value of the robustness sign at this moment. The robustness sign is the value at the previous time step. It is the back electromotive force. This is the value of the back electromotive force at the previous moment.

[0022] make

[0023] Furthermore, the input to the equivalent position error module 2-3 is the output of the gradient search module 2-2 (q). α q β Its output is the robustness sign (q) α q β The equivalent positional error (q) formed on the γ and δ axes errγ q errδ The calculation process is as follows:

[0024]

[0025] in, This indicates that the phase-locked loop output estimates the rotor position.

[0026] Furthermore, the input to the cost function selection module 2-4 is the output of the equivalent position error module 2-3 (q). errγ q errδ First, two different cost functions are defined to determine the iteration direction. Then, two cost functions J1 and J2 are constructed by combining the equivalent position error. The calculation process is as follows:

[0027]

[0028] Where θ err This represents the error between the actual rotor position and the output rotor position, with a value of:

[0029] Furthermore, based on the cost function equations (J1, J2) selected by the cost function selection module 2-4, and combined with the iterative equations of the iterative module 2-5, the estimated rotor position angle is obtained. Perform three iterations to output the final estimated rotor position angle. The calculation process is as follows:

[0030]

[0031] in This is the estimated rotor position result from the final iteration. When q errδ When ≥0, only θ err =0

[0032] J1 = 0, meaning J1 is chosen as the cost function within the interval (-π / 2, π / 2), and the angle error (θ) err After three iterations, it will converge to 0. When q errδ When <0, at θ err = -π and θ err When π, J2 = 0, meaning that in the interval (-π, -π / 2) ∪ (π / 2, π), J2 is chosen as the cost function, and the angle error (θ) err After three iterations, it will converge to π or -π, therefore, an accurate estimate of the angle needs to be based on q. errγ The positive or negative value is used to determine the output result. Add or subtract π. Input the accurate estimated angle into the differential module 2-6, and differentiate it to obtain the estimated electric speed of the motor. The calculation process is as follows:

[0033]

[0034] further, Figure 4 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The experimental results of an improved anti-interference prediction phase-locked loop (PLL) in the PLL of the permanent magnet synchronous motor without a position control system under static conditions at a low speed of 150 r / min are also presented. Figure 4 It can be seen that when the permanent magnet synchronous motor is running at a constant speed, θ err Within ±12°, the rotational speed error is small.

[0035] further, Figure 5 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that an improved anti-interference predictive phase-locked loop (PLL) was used in the PLL of the permanent magnet synchronous motor without a position control system under acceleration conditions where the speed increased from 1000 r / min to 2000 r / min. Figure 5 It can be seen that, regardless of whether the motor is running at a constant speed or accelerating, θ err All within ±5°, with the maximum speed error during acceleration being only 25 r / min.

[0036] further, Figure 6 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that the motor, operating at a constant speed of 2000 r / min, is subjected to a sudden application of a rated load of 2.3 Nm at 2 seconds using an improved anti-interference predictive PLL in its position-uncontrolled PLL. Based on... Figure 6 It can be seen that under the condition of sudden load increase, θ err The maximum angular error is only between -8° and 10°, and the angular error θ is lower after the rotational speed stabilizes. errWithin ±5°, the maximum speed error is around -100 r / min.

[0037] further, Figure 7 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that an improved anti-interference prediction PLL was used in the PLL of the permanent magnet synchronous motor without a position control system under sudden changes in speed and load. Under constant speed of 1000 r / min, the rated load of 2.3 Nm was suddenly applied at 1.25 s, then reduced to half load of 1.15 Nm at 2.5 s, then the speed suddenly increased to 2000 r / min at 3.5 s, the load suddenly decreased to 0 Nm at 4.5 s, and finally the speed suddenly decreased to 1000 r / min at 5.5 s. Based on... Figure 7 It can be seen that under complex changing operating conditions, θ err The maximum fluctuation in angle error is only between -15° and 15°, while the maximum fluctuation in speed error is between -120 r / min and 80 r / min. After the speed stabilizes, the angle error θ... err Within ±5°, the rotational speed error is approximately -20 r / min to 20 r / min.

[0038] Furthermore, the experimental results above demonstrate that the sensorless control method for permanent magnet synchronous motors based on an improved anti-interference predictive phase-locked loop can achieve high-performance control of permanent magnet synchronous motors and obtain satisfactory control results. Attached Figure Description

[0039] The accompanying drawings, as part of this invention, are provided to further illustrate the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention, but do not constitute an undue limitation thereof. Clearly, the drawings described below are merely some embodiments, and those skilled in the art can obtain other drawings based on these drawings without any creative effort.

[0040] Figure 1 This is a schematic diagram of a permanent magnet synchronous motor control system provided in an embodiment of the present invention;

[0041] Figure 2 This is an equivalent position error vector diagram provided in one embodiment of the present invention;

[0042] Figure 3 This refers to the partitioning of the entire computational domain and the selection of the cost function provided in one embodiment of the present invention;

[0043] Figure 4 This is an experimental result of the permanent magnet synchronous motor speed and rotor position estimation system provided by the present invention under low-speed operating conditions in an embodiment of the present invention.

[0044] Figure 5 This is an experimental result of the permanent magnet synchronous motor speed and rotor position estimation system provided by the present invention under the forward acceleration condition of a permanent magnet synchronous motor according to an embodiment of the present invention;

[0045] Figure 6 This is an experimental result of the permanent magnet synchronous motor speed and rotor position estimation system provided by the present invention under the condition of sudden full load on a permanent magnet synchronous motor according to an embodiment of the present invention;

[0046] Figure 7 The present invention provides experimental results of the permanent magnet synchronous motor speed and rotor position estimation system under complex operating conditions of sudden changes in speed and load of the permanent magnet synchronous motor.

[0047] It should be noted that these accompanying drawings and textual descriptions are not intended to limit the scope of the invention in any way, but rather to illustrate the concept of the invention to those skilled in the art by referring to specific embodiments. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments will be clearly and completely described below with reference to the accompanying drawings. The following embodiments are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0049] To enhance understanding of the present invention, the technical solution of the present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0050] like Figure 1 As shown, the sensorless control method for permanent magnet synchronous motors based on an improved anti-interference predictive phase-locked loop specifically includes the following steps:

[0051] Step S100: The phase detector module 1 is used to identify the relationship between the back electromotive force information and the rotor position. Its input is the motor stator current (i) in the two-phase stationary coordinate system. α i β ), motor stator voltage in two-phase stationary coordinate system (u) α u β ).

[0052] Step S200: This invention uses a first-order sliding mode observer to observe the back electromotive force of a permanent magnet synchronous motor, and its output is the back electromotive force.

[0053] Specifically, the calculation process is as follows:

[0054]

[0055] in Estimate the current for the α-axis. To estimate the current along the β axis, L d L q These are the d-axis and q-axis inductances (Wb) of the permanent magnet synchronous motor, respectively. The electric angular velocity is an estimated value, with the symbol "·" representing the differential operator and "^" representing the estimated value. The sliding mode equivalent control function is designed, and its calculation process is as follows:

[0056]

[0057] Where z α and z β This is the control input of the SMO, where `sign` is the sign function and `k` is the gain of the first-order sliding mode observer. The back EMF signal is then filtered by the LPF. The calculation process is as follows:

[0058]

[0059] These are the estimated back electromotive forces along the α-axis and β-axis, respectively. To estimate the difference between the α-axis current and the measured value To estimate the difference between the β-axis current and the measured value k is the sliding mode gain, ω c This is the cutoff frequency of the low-pass filter.

[0060] Step S300: In this embodiment of the application, the robustness symbol q α and q β Obtain it through the following steps:

[0061] Step S310: Integrate the back electromotive force information to obtain the estimated rotor flux linkage value.

[0062] Specifically, the calculation process is as follows:

[0063]

[0064] in, This indicates the output estimate of the rotor flux linkage, ψ. f For permanent magnet flux linkage, A α A β Estimated back electromotive force along the α and β axes respectively The lumped interference, θ e Indicates the actual rotor position.

[0065] Step S320: A novel dual-cost function prediction phase-locked loop strategy based on gradient descent is proposed, which combines interference suppression techniques to enhance the robustness of the system and address these limitations.

[0066] Specifically, we first establish a linear regression equation based on gradient descent and construct the gradient function. The calculation process is as follows:

[0067]

[0068] The corrected robust signal is represented as q α and q β The error is minimized by using the partial derivatives of the gradient equation to form a compensation phase. The calculation process is as follows:

[0069]

[0070] Step S330: Reduce the error between the input waveform and the theoretical value using a correction term obtained through a gradient function. The corrected robust signal is denoted as q. α and q β The calculation process is as follows:

[0071]

[0072] Where μ is the gain of the gradient search, The value of the robustness sign at this moment. The robustness sign is the value at the previous time step. It is the back electromotive force. This is the value of the back electromotive force at the previous moment.

[0073] make

[0074] Step S400, as follows Figure 2 As shown, the output is the robustness sign (q). α q β The equivalent positional error (q) formed on the γ and δ axes errγ q errδ The calculation process is as follows:

[0075]

[0076] in, This indicates that the phase-locked loop output estimates the rotor position.

[0077] Furthermore, the input to the cost function selection module 2-4 is the output of the equivalent position error module 2-3 (q). errγ q errδ First, two different cost functions are defined to determine the iteration direction. Then, two cost functions J1 and J2 are constructed by combining the equivalent position error. The calculation process is as follows:

[0078]

[0079] Where θe θ represents the actual rotor position. err This represents the error between the actual rotor position and the output rotor position, with a value of:

[0080] Furthermore, based on the cost function equations (J1, J2) selected by the cost function selection module 2-4, and combined with the iterative equations of the iterative module 2-5, the estimated rotor position angle is obtained. Perform three iterations to output the final estimated rotor position angle. The calculation process is as follows:

[0081] Furthermore, the two cost functions are expanded using a first-order Taylor series, and the calculation process is as follows:

[0082]

[0083] Where i is the iteration number. It is the output value of the (i-1)th iteration, and is also used as the input value of the i-th iteration. This is the final output value of the iteration. Higher-order infinitesimals are ignored. when and When the value approaches zero, the iterative formula for the Newton-Raphson iterative method can be obtained, and its calculation process is as follows:

[0084]

[0085] in This is the estimated rotor position result from the final iteration. For example... Figure 3 As shown, when q errδ When ≥0, only θ err When θ = 0, J1 = 0, meaning that J1 is chosen as the cost function in the interval (-π / 2, π / 2), and the angle error (θ) err After three iterations, it will converge to 0. When q errδ When <0, at θ err = -π and θ err When π, J2 = 0, meaning that in the interval (-π, -π / 2) ∪ (π / 2, π), J2 is chosen as the cost function, and the angle error (θ) err After three iterations, it will converge to π or -π, therefore, an accurate estimate of the angle needs to be based on q. errγ The positive or negative value is used to determine the output result. Add or subtract π. Input the accurate estimated angle into the differential module 2-6, and differentiate it to obtain the estimated electric speed of the motor. The calculation process is as follows:

[0086]

[0087] further, Figure 4 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The experimental results of an improved anti-interference prediction phase-locked loop (PLL) in the PLL of the permanent magnet synchronous motor without a position control system under static conditions at a low speed of 150 r / min are also presented. Figure 4 It can be seen that when the permanent magnet synchronous motor is running at a constant speed, θ err Within ±12°, the rotational speed error is small.

[0088] further, Figure 5 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that an improved anti-interference predictive phase-locked loop (PLL) was used in the PLL of the permanent magnet synchronous motor without a position control system under acceleration conditions where the speed increased from 1000 r / min to 2000 r / min. Figure 5 It can be seen that, regardless of whether the motor is running at a constant speed or accelerating, θ err All within ±5°, with the maximum speed error during acceleration being only 25 r / min.

[0089] further, Figure 6 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that the motor, operating at a constant speed of 2000 r / min, is subjected to a sudden application of a rated load of 2.3 Nm at 2 seconds using an improved anti-interference predictive PLL in its position-uncontrolled PLL. Based on... Figure 6 It can be seen that under the condition of sudden load increase, θ err The maximum angular error is only between -8° and 10°, and the angular error θ is lower after the rotational speed stabilizes. err Within ±5°, the maximum speed error is around -100 r / min.

[0090] further, Figure 7 Experimental results of a five-pole logarithmic permanent magnet synchronous motor with a rated speed of 3000 r / min are presented. The results show that an improved anti-interference prediction PLL was used in the PLL of the permanent magnet synchronous motor without a position control system under sudden changes in speed and load. Under constant speed of 1000 r / min, the rated load of 2.3 Nm was suddenly applied at 1.25 s, then reduced to half load of 1.15 Nm at 2.5 s, then the speed suddenly increased to 2000 r / min at 3.5 s, the load suddenly decreased to 0 Nm at 4.5 s, and finally the speed suddenly decreased to 1000 r / min at 5.5 s. Based on... Figure 7 It can be seen that under complex changing operating conditions, θ err The maximum fluctuation in angle error is only between -15° and 15°, while the maximum fluctuation in speed error is between -120 r / min and 80 r / min. After the speed stabilizes, the angle error θ... errWithin ±5°, the rotational speed error is approximately -20 r / min to 20 r / min.

[0091] Furthermore, the experimental results above demonstrate that the sensorless control method for permanent magnet synchronous motors based on an improved anti-interference predictive phase-locked loop can achieve high-performance control of permanent magnet synchronous motors and obtain satisfactory control results.

Claims

1. This invention provides a sensorless control method for permanent magnet synchronous motors based on an improved anti-interference predictive phase-locked loop, characterized in that, The sensorless control method for permanent magnet synchronous motors based on an improved anti-interference predictive phase-locked loop includes a phase detector module 1 and an improved anti-interference predictive phase-locked loop module 2. The improved anti-interference predictive phase-locked loop module 2 includes an integration module 2-1, a gradient search module 2-2, an equivalent error cost function selection module 2-3, an iteration module 2-4, and a differentiation module 2-5.

2. The sensorless control method for a permanent magnet synchronous motor according to claim 1, characterized in that: The back EMF estimation uses a first-order sliding mode observer to observe the back EMF of the permanent magnet synchronous motor, and its output is the back EMF. The calculation process is as follows: Among them, Estimate the current for the α-axis. To estimate the current along the β axis, L d L q These are the d-axis and q-axis inductances (Wb) of the permanent magnet synchronous motor, respectively. The electric angular velocity is an estimated value; the symbol "·" represents the differential operator; the symbol "^" represents the estimated value. The sliding mode equivalent control function is designed, and its calculation process is as follows: Where z α and z β is the control input of the SMO, sign is the sign function, and k is the gain of the first-order sliding mode observer.

3. The sensorless control method for a permanent magnet synchronous motor according to claim 2, characterized in that: Back EMF value after LPF filtering for: in, These are the estimated back electromotive forces along the α-axis and β-axis, respectively. To estimate the difference between the α-axis current and the measured value To estimate the difference between the β-axis current and the measured value k is the sliding mode gain, ω c This is the cutoff frequency of the low-pass filter.

4. The anti-interference predictive phase-locked loop according to claim 1, characterized in that: The improved anti-interference prediction phase-locked loop module 2 includes an integration module 2-1, a gradient search module 2-2, an equivalent position error module 2-3, a cost function selection module 2-4, an iteration module 2-5, and a differentiation module 2-6.

5. The sensorless control method for a permanent magnet synchronous motor according to claim 4, characterized in that: The integration module 2-1 is used to integrate the back electromotive force information to obtain the estimated rotor flux linkage value. Its output is the input of gradient search module 2-2. The calculation process is as follows: in, This indicates the output estimate of the rotor flux linkage, ψ. f For permanent magnet flux linkage, A α A β Estimated back electromotive force along the α and β axes respectively The lumped interference, θ e Indicates the actual rotor position.

6. The sensorless control method for a permanent magnet synchronous motor according to claim 5, characterized in that: The output q of the gradient search module 2-2 α q β This is the robustness sign. Its calculation process is as follows: Where μ is the gain of the gradient search, The value of the robustness sign at this moment. The robustness sign is the value at the previous time step. It is the back electromotive force. This is the value of the back electromotive force at the previous moment. make 7. The sensorless control method for a permanent magnet synchronous motor according to claim 6, characterized in that: The output of the equivalent position error module 2-3 is the robustness sign q. α q β The equivalent position error q formed on the γ and δ axes errγ q errδ The calculation process is as follows: in, This indicates that the phase-locked loop output estimates the rotor position.

8. The sensorless control method for a permanent magnet synchronous motor according to claim 7, characterized in that: The cost function selection module 2-4 defines two different cost functions to determine the iteration direction. Combining the equivalent position error, two cost functions J1 and J2 are constructed, and their calculation process is as follows: Where θ err This represents the error between the actual rotor position and the output rotor position, with a value of:

9. The sensorless control method for a permanent magnet synchronous motor according to claim 8, characterized in that: Based on the cost function equations J1 and J2 selected by the cost function selection module 2-4, and combined with the iterative equations of the iterative module 2-5, the estimated rotor position angle is obtained. Perform three iterations to output the final estimated rotor position angle. The calculation process is as follows: in This is the estimated rotor position result from the final iteration. When q errδ When ≥0, only θ err When θ = 0, J1 = 0, meaning that J1 is chosen as the cost function in the interval (-π / 2, π / 2), and the angle error (θ) err After three iterations, it will converge to 0. When q errδ When <0, at θ err = -π and θ err When π, J2 = 0, meaning that in the interval (-π, -π / 2) ∪ (π / 2, π), J2 is chosen as the cost function, and the angle error (θ) err After three iterations, it will converge to π or -π, therefore, an accurate estimate of the angle needs to be based on q. errγ The positive or negative value is used to determine the output result. Add or subtract π. Input the accurate estimated angle into the differential module 2-6, and differentiate it to obtain the estimated electric speed of the motor. The calculation process is as follows:

Images