A method for automatic balancing of an electromagnetic bearing-rotor system without rotational speed

Abstract

The application discloses a kind of without speed electromagnetic bearing-rotor system automatic balancing method, this method first establishes the mathematical model of linear extended state observer (LESO), then designs the control structure of linear extended state observer wave trap, further carries out linear extended state observer pole configuration and inlay physical model, finally introduces Fal nonlinear function.The application does not need real-time speed information, can effectively reduce the same frequency vibration of electromagnetic bearing-rotor system, and significantly reduces control current and base vibration.

Description

Technical Field

[0001] This invention relates to the field of vibration control of electromagnetic bearing-rotor systems, and specifically to an automatic balancing method for electromagnetic bearing-flexible rotor systems that does not require rotational speed. Background Technology

[0002] Electromagnetic bearing-rotor systems are widely used in high-speed rotating machinery. Due to machining errors, material inhomogeneity, and other reasons, rotors inevitably exhibit imbalance, generating an unbalanced excitation force synchronized with the rotational speed during rotation. The amplitude of this unbalanced excitation force is proportional to the square of the rotational speed. At high speeds, this unbalanced force not only causes rotor vibration but also transmits it through the bearings to the base, causing base vibration. Simultaneously, the increased control current can easily lead to power amplifier saturation, affecting the stability of the electromagnetic bearing system. Currently, the solution to this problem is to use a notch filter to remove the same-frequency component from the displacement signal, thereby reducing the control current and causing the rotor to tend to rotate around its axis of inertia. Such an electromagnetic bearing system is called an automatic balancing system.

[0003] Currently, the main control methods for electromagnetic bearing-rotor systems include control structures using cascaded general-purpose filters and structures employing insertion-type adaptive filters. Because general-purpose notch filters exhibit phase lag for signals before the center frequency, they are mostly used in constant-speed applications. For variable-speed applications, an adaptive least mean square algorithm is generally used. This algorithm constructs a rotating coordinate system based on the real-time rotational speed, transforming the sinusoidal vibrations in the x and y directions into fixed points within the rotating coordinate system. Then, an integrator is used to track the signal and apply notch filtering.

[0004] Existing technologies heavily rely on real-time rotational speed, which presents the following problems: (1) Additional rotational speed sensors increase system cost and reduce system reliability. In embedded digital controllers, there is a contradiction between measurement accuracy and real-time performance in pulse timing. When the motor speed increases rapidly or when the motor stops under load, inaccurate rotational speed is likely to occur, which in turn affects system stability; (2) Generalized integrators and least mean square algorithms with iterative processes present a contradiction between speed and stability, iteration step size and convergence speed. In actual operation, repeated iterations and inconsistent results may occur; (3) Notch filters affect system phase and can cause system instability before the rigid body's critical rotational speed. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of existing technologies and provide an automatic balancing method for magnetic bearing-rotor systems that does not require a speed signal. A linear extended state observer (LESO) is used to construct a notch filter. This method can reduce vibration, control current, and base vibration of the electromagnetic bearing-rotor system without requiring real-time speed.

[0006] The objective of this invention is achieved through the following technical solution: an automatic balancing method for an electromagnetic bearing-rotor system without the need for rotational speed, the method comprising the following steps:

[0007] S1, Establish the mathematical model of the linear extended state observer, specifically as follows:

[0008] The single control channel of the electromagnetic bearing-rotor system is simplified into a second-order system, described as follows:

[0009]

[0010] Where y is displacement, ξ is disturbance including unbalanced force of the same frequency, bu is control force generated by controller, λ1, λ2 and b represent the characteristics of second-order object, and b is composed of known part b0 and unknown part;

[0011] When designing the controller, the total disturbance is defined as:

[0012]

[0013] Treating the total disturbance as the third state variable, the state equation of the linear extended state observer constructed for the controlled second-order object and the disturbance is:

[0014]

[0015] Wherein, the state vector z = [z1 z2 z3] T z1, z2 and z3 are the observed values ​​of displacement, displacement derivative and total disturbance of electromagnetic bearing-rotor system, respectively. A is the system matrix of the observer, B is the input matrix, C is the output matrix and L is the correction matrix.

[0016] When the equation of state does not contain a physical model C = [1 0 0], Where β1, β2, and β3 are undetermined coefficients obtained by configuring the observer poles;

[0017] When the equation of state includes a physical model C = [1 0 0], Where a0 is the observer stiffness and a1 is the observer damping;

[0018] S2, Design the control structure of the linear extended state observer notch filter to attenuate the vibration caused by the disturbance, specifically:

[0019] The input signal of the linear extended state observer is set as displacement y, and the output signal of the linear extended state observer is set as state variable z1. First, in the first discrete cycle, the input signal of the linear extended state observer is set as displacement y(1). Then, after one discrete cycle, the output signal is z1(1). In the second discrete cycle, the current displacement y(2) is subtracted from the output z1(1), and the result is used as the new input of the linear extended state observer. After one discrete cycle, the output signal is z1(2), which is used to subtract z1(2) from y(3) in the third discrete cycle. This process is repeated, and the signal after each subtraction is fed into the controller to maintain the stability of the system.

[0020] S3, perform pole placement for the linearly extended state observer. The specific pole placement method is as follows:

[0021] The poles of the linear extended state observer are divided into a pair of conjugate poles s1 and s2 that determine the resonant frequency and damping ratio, and a pole s3 that determines the tracking speed; the conjugate poles s1 and s2 are in the following form:

[0022]

[0023] Where η is the damping ratio, ω n is the resonant frequency, and j is the imaginary unit; this pair of poles causes the linear extended state observer to produce a resonant region at the resonant frequency;

[0024] When the input of the linear extended state observer is displacement y and the output is the first state variable z1 of the state equation, within the frequency range below the resonance region, the output of the linear extended state observer is equal to the input; within the resonance region, the output of the linear extended state observer first increases and then decreases; within the frequency range above the resonance region, the output of the linear extended state observer is less than the input; the resonance frequency ω... n The damping ratio η is used to adjust the center position of the resonant region, and the damping ratio η is used to adjust the resonant region damping of the linear extended state observer.

[0025] The third pole s3 has the following form:

[0026] s3=-eηω n

[0027] Where e is the error amplification ratio, used to reduce the tracking error of the linear extended state observer;

[0028] When the frequency of the input signal exceeds the resonant frequency of the linear extended state observer, the output signal of the linear extended state observer cannot accurately track the input signal, and its output tends to decrease. When the pole s3 is placed further away in the left half-plane, the resonant frequency is not changed, but the output amplitude is increased, and the tracking error is reduced.

[0029] S4, Embedded Physical Model

[0030] The physical model can be embedded in the state equation of the linear extended state observer. The observer stiffness a0 is used to adjust the resonant frequency. Increasing the value of the observer stiffness a0 increases the resonant frequency, which has the same effect as increasing the resonant frequency of the poles. The observer damping a1 is used to adjust the resonance effect of the linear extended state observer. Decreasing the observer damping a1 enhances the negative gain at the resonant point, which has the same effect as decreasing the damping ratio η of the pole placement. By embedding the physical model, the value of the L matrix of the observer's mathematical model is reduced, thereby reducing the noise sensitivity.

[0031] Furthermore, the transfer function from the displacement input y to the running output z1 of the linear extended state observer is:

[0032]

[0033] Where s is the Laplace operator; the transfer function of the notch filter is:

[0034]

[0035] Due to the structure of the notch filter, the notch filter has a negative gain of at least 6 dB in the range below the observer's resonant region.

[0036] Furthermore, the effects of the pole placement method on the frequency characteristics of the notch filter of the linear extended state observer are as follows:

[0037] The conjugate poles that determine the resonant frequency and damping ratio create a resonant region for the observer near the resonant frequency. Within this region, the amplitude of the notch filter first decreases and then increases with frequency, forming a negative gain region with a minimum value and a positive gain region with a maximum value. The frequency of the minimum value in the negative gain region is equal to the resonant frequency. The smaller the damping ratio, the more the minimum and maximum values ​​in the resonant region increase, the wider the resonant region becomes, and the wider the frequency band of the negative gain region within the resonant region becomes.

[0038] The poles that determine the tracking speed affect the maximum value of the positive gain region in the resonant region; when the error amplification ratio is in the range of 0 to 5, the larger the error amplification ratio, the smaller the maximum value of the positive gain region in the resonant region; the phase characteristic in the resonant region starts from 0 at the resonant frequency and increases, then decreases back to 0.

[0039] Furthermore, when the frequency of the input signal is less than the resonant region, the notch filter has a base gain of -6dB, and the gain decreases in a frequency range before the resonant frequency. Within the resonant region of the notch filter in the linear extended state observer, the amplitude-frequency characteristic forms a notch region, which acts as a notch filter. After the frequency exceeds the notch region, the gain of the notch filter gradually increases to 0 and loses its function.

[0040] Furthermore, when the frequency of the input signal is below the resonant frequency, the phase characteristic is 0. However, after exceeding the observer bandwidth, the phase gradually increases and eventually returns to 0. Therefore, the linear extended state observer notch filter only reduces the gain in the frequency band below the observer bandwidth and in the notch region near the bandwidth, and does not affect the phase of the controller.

[0041] Furthermore, the resonant frequency ω n The system is set to its rated operating frequency. The system will have optimal current suppression during rated operation and will also have suppression before accelerating to the rated speed.

[0042] Furthermore, the linear extended state observer has a finite gain for the input signal, the notch filter of the linear extended state observer has a finite gain at the resonant frequency, and the notch depth is adjustable.

[0043] Furthermore, the output of the linear extended state observer is followed by a Fal nonlinear function to divide the total gap into a strong control region and a weak control region for automatic balancing. The criterion for the boundary between the two control regions is 30% of the total gap. The expression for the Fal function is:

[0044]

[0045] Where p is the input to the Fal function; δ is the linear domain, which is set as the strong control region for automatic balancing; α is the nonlinearity factor, which can be reduced to enhance the nonlinearity.

[0046] The stronger the nonlinearity, the smaller the output in the weak control region. After applying the Fal function, for vibration signals exceeding the strong control region, an automatic attenuation and balancing output method is adopted to avoid excessive vibration and rubbing against the protective bearing.

[0047] Furthermore, the method does not require real-time speed information, can effectively reduce the vibration of the electromagnetic bearing-rotor system, and significantly reduce control current and base vibration, without requiring shutdown operation near the critical speed range.

[0048] The basic principle of this invention is as follows: During the operation of an active electromagnetic bearing-rotor system, the unbalanced excitation force with the same frequency as the rotational speed causes an increase in control current and vibration of the rotor and base, which must be suppressed. By establishing a mathematical model of a linear extended state observer, constructing a control structure for the notch filter of the linear extended state observer, performing pole placement of the linear extended state observer, embedding a physical model, and using the Fal nonlinear function, an automatic balancing method for an electromagnetic bearing-rotor system without requiring a rotational speed signal is proposed. The input of the linear extended state observer is set as the rotor displacement, and the output is set as the observer state variable z1. Then, a notch filter structure is used to weaken the input displacement using the state variable z1. Adjusting the pole placement of the linear extended state observer changes the performance of the notch filter, so that the notch filter has negative gain both near and below the resonant frequency, thereby weakening the signal in the displacement that has the same frequency as the rotational speed, forming an automatic balancing system.

[0049] The advantages of this invention compared to the prior art are:

[0050] 1. No speed signal is required, and it can be directly applied to existing industrial magnetic bearing control systems;

[0051] 2. The notch filter mentioned in the content does not change the phase characteristics of the system and will not affect the stability of the system. Attached Figure Description

[0052] Figure 1 This is a flowchart of the present invention;

[0053] Figure 2 This is a block diagram of the linear extended state observer notch filter of the present invention;

[0054] Figure 3 Bode plot of the linear extended state observer notch filter with different damping ratios;

[0055] Figure 4 Bode plot of the linear extended state observer notch filter when configuring different error amplification ratios;

[0056] Figure 5 The amplitude-frequency response of the linear extended state observer notch filter when configuring different embedded physical model information;

[0057] Figure 6 This is a graph showing the input and output characteristics of the Fal function. Detailed Implementation

[0058] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0059] like Figure 1As shown, the implementation process of an automatic balancing method for an electromagnetic bearing-rotor system without the need for a speed signal is as follows: First, a mathematical model of the Linear Extended State Observer (LESO) is established; then, the notch filter structure of the LESO is designed; next, the pole placement and embedded physical model of the LESO are performed; and finally, the Fal nonlinear function is introduced.

[0060] Step (1) Establish the mathematical model of the linear extended state observer

[0061] A single control channel of an electromagnetic bearing-rotor system can be simplified to a second-order system, described by the following differential equation:

[0062]

[0063] Where y is displacement, ξ is disturbance including unbalanced force of the same frequency, bu is control force generated by the controller, u is output of the controller, λ1, λ2 and b represent the characteristics of the second-order object, b consists of known and unknown parts, where the known part is b0.

[0064] When designing the controller, because the characteristics of the second-order object are unknown, the perturbation is defined as...

[0065]

[0066] The disturbance includes the characteristics of the second-order object and the external perturbation force ξ, and is called the total disturbance. To observe the total disturbance, it is treated as a third state variable. The state equation of the linear extended state observer constructed for the controlled second-order object and the disturbance is:

[0067]

[0068] Wherein, the state vector z = [z1 z2 z3] T z1, z2, and z3 are the observed values ​​of displacement, derivative of displacement, and total disturbance of the electromagnetic bearing-rotor system, respectively. Let A be the output of the observer, B be the system matrix of the state observer, C be the input matrix, and L be the correction matrix. The superscript T of the matrix indicates transpose.

[0069] When the state equation of the linear extended state observer does not contain a physical model, the parameters in the state equation of the linear extended state observer are as follows: C = [1 0 0], β1, β2, and β3 are undetermined coefficients obtained by configuring the observer poles. When the equations include a physical model, the parameters in the state equations of the linear extended state observer are as follows: C = [1 0 0], a0 is the observer stiffness, and a1 is the observer damping.

[0070] Step (2) Design the control structure of the linear extended state observer notch filter.

[0071] Automatic balancing of the electromagnetic bearing-rotor system is achieved by weakening the displacement signal with the same frequency as the rotational speed in the controller. Therefore, a notch filter is used to weaken the same-frequency signal in the displacement sensor, thereby reducing the control current and the force transmitted to the base. The disturbance is essentially the difference between the displacement y and the state variable z1 in the observer; therefore, the observer state variable z1 represents the cumulative effect of the total disturbance up to the current moment. The vibration caused by the disturbance is attenuated using the following control structure of a linear extended state observer notch filter: In the linear extended state observer notch filter structure, the input signal of the linear extended state observer is set as the displacement y, and the output signal of the linear extended state observer is set as the state variable z1. First, in the first discrete cycle, the input signal of the linear extended state observer is set to the displacement y(1). Then, after one discrete cycle, the output signal of the linear extended state observer is z1(1). In the second discrete cycle, the current displacement y(2) is subtracted from the output z1(1) of the linear extended state observer, and the result is used as the new input of the linear extended state observer. After one discrete cycle, the linear extended state observer generates an output signal z1(2), which is used to subtract z1(2) from y(3) in the third discrete cycle. This process continues in this manner. At the same time, the signal after each subtraction is fed into the controller to maintain the stability of the system. The structure of the linear extended state observer notch filter and the controller is as follows: Figure 2 As shown.

[0072] Step (3) Perform pole placement for the linear extended state observer.

[0073] The classic linear extended state observer pole placement method places poles at the same location, called the observer bandwidth. However, the notch filter of the linear extended state observer configured using this method has a negative gain of only -7dB near the observer bandwidth, which cannot meet the requirements of automatic balancing. Therefore, this invention proposes a new method to address the pole placement problem.

[0074] Based on the pole principle of mechanical vibration, this invention divides the poles of the linear extended state observer into a pair of conjugate poles s1 and s2 that determine the resonant frequency and damping ratio, and a pole s3 that determines the tracking speed. s1 and s2 are written in the following form.

[0075]

[0076] Where η is the damping ratio, ω n The resonant frequency is denoted by . This pair of poles causes the linear extended state observer to produce a resonant region at the resonant frequency.

[0077] When the input of the linear extended state observer is displacement y, and the output of the linear extended state observer is the first state variable z1 of the state equation, the output of the linear extended state observer is equal to the input in the frequency range below the resonance region; in the resonance region, the output of the linear extended state observer first increases and then decreases; in the frequency range above the resonance region, the output of the linear extended state observer is less than the input. Resonant frequency ω n Used to adjust the center position of the resonant region. The damping ratio η is used to adjust the resonant region damping of the linear extended state observer.

[0078] The transfer function from the input to the output of the linear extended state observer is:

[0079]

[0080] Where s is the Laplace operator. The transfer function of the notch filter is:

[0081]

[0082] Due to the notch filter's structure, it exhibits a negative gain of at least 6 dB (1 / 2 times) below the observer's resonant region. The placement of the poles determines the three coefficient terms in the transfer function. With the resonant frequency set to 400 Hz, the Bode plots of the notch filter's transfer function at different damping ratios are shown below. Figure 3 As shown. By Figure 3 It can be seen that the smaller the damping ratio, the smaller the gain of the linear extended state observer notch filter at the resonant frequency, and the more obvious the notch effect of the linear extended state observer notch filter.

[0083] The third pole is configured as follows

[0084] s3=-eηω n

[0085] Where e is the error amplification ratio, used to reduce the tracking error of the linear extended state observer.

[0086] When the frequency of the input signal exceeds the resonant frequency of the linear extended state observer, the output signal of the linear extended state observer cannot accurately track the input signal, and its output tends to decrease. Placing the pole s3 further away in the left half-plane does not change the resonant frequency, but increases the output amplitude and reduces the tracking error. The Bode plots of the notch filter transfer function under different error amplification ratios are shown below. Figure 4 As shown. By Figure 4 It can be seen that when the error amplification ratio is 3, the peak value of the notch filter of the linear extended state observer decreases, thus reducing high-frequency noise.

[0087] The effects of this pole placement method on the frequency characteristics of the linear extended state observer notch filter are as follows: The conjugate poles determining the resonant frequency and damping ratio create a resonant region near the resonant frequency. Within this region, the amplitude of the notch filter first decreases and then increases with frequency, forming a negative gain region with a minimum value and a positive gain region with a maximum value. The frequency of the minimum value in the negative gain region is equal to the resonant frequency. The smaller the damping ratio, the higher both the minimum and maximum values ​​within the resonant region, widening the resonant region and the frequency band of the negative gain region within it. Based on -6dB, the gain of this notch filter decreases in a frequency range before the resonant frequency, reaching its lowest point at the resonant frequency, and then gradually increases. The smaller the damping ratio, the stronger the resonance and anti-resonance effects, the wider the negative gain region, and the lower the minimum gain. With a damping ratio of 0.075, a gain of -18dB can be achieved at the resonant frequency, while a large negative gain exists in the 250Hz–400Hz range before the resonant frequency. The maximum negative gain can be adjusted by regulating the damping ratio, thereby regulating the notch depth. Therefore, setting the resonant frequency to the system's rated operating frequency will provide optimal current suppression during rated operation, as well as suppression before accelerating to the rated speed. Furthermore, due to the -6dB negative gain in the low-frequency range, the controller's gain parameter should be appropriately increased to maintain system stability.

[0088] The poles that determine the tracking speed affect the maximum value of the positive gain region in the resonant region. When the error amplification ratio is in the range of 0 to 5, the larger the error amplification ratio, the smaller the maximum value of the positive gain region in the resonant region. The phase characteristic in the resonant region starts from 0 at the resonant frequency, increases, and then decreases back to 0. Therefore, the linear extended state observer notch filter only reduces the gain in the frequency band below the observer bandwidth and in the notch region near the bandwidth, without affecting the phase of the controller, thus avoiding any impact on the stability of the original system.

[0089] After pole placement, the parameters β1, β2, and β3 of the correction matrix L can be obtained.

[0090] Step (4) Embedding the physical model

[0091] Setting the second-order physical model parameters in the observer's state equation can adjust the resonance effect of the linear extended state observer notch filter. The effects of different observer stiffness a0 and observer damping a1 on the resonator's frequency characteristics are as follows: Figure 5As shown in the diagram, the observer stiffness a0 is used to adjust the resonant frequency. Increasing the model stiffness raises the resonant frequency, similar to increasing the pole resonant frequency. The observer damping a1 is used to adjust the resonance effect of the linearly extended state observer. Adding negative damping to the model enhances the negative gain at the resonant point, similar to decreasing the pole damping ratio. Directly configuring a large resonant frequency and a small damping ratio would result in an excessively large L-matrix value, amplifying noise. By embedding the physical model, the L-matrix value of the observer's mathematical model can be reduced, thereby reducing noise sensitivity.

[0092] Step (5) Introduce the Fal nonlinear function

[0093] The output of the linear extended state observer is followed by a Fal nonlinear function. The total gap is divided into a strong control region and a weak control region for automatic balancing, with the boundary between the two control regions determined by 30% of the total gap. Within the strong control region, the output of the Fal function should equal the input. Therefore, its expression is:

[0094]

[0095] Where p is the input to the Fal function; δ is the linear domain, which is set as the strong control region for automatic balancing; α is the nonlinearity factor, which can be reduced to enhance the nonlinearity.

[0096] The stronger the nonlinearity, the smaller the output in the weak control region. After applying the Fal function, for vibration signals exceeding the strong control region, an automatic output balancing method is adopted to prevent excessive vibration from rubbing against the protective bearing. The input / output characteristic graph of the Fal nonlinear function is shown below. Figure 6 As shown.

[0097] The above steps constitute an automatic balancing method for an electromagnetic bearing-rotor system that does not require a speed signal. This method significantly reduces vibration and current in the non-critical speed range, reduces the transmitted force on the base, and thus reduces base vibration. It does not affect the stability near the critical speed range and can be directly applied to industrial magnetic bearing control systems without the need for additional speed sensors. Contents not described in detail in this specification are prior art known to those skilled in the art.

[0098] The above examples are used to illustrate the present invention, not to limit it. Any modifications and changes made to the present invention within the spirit and scope of the claims fall within the protection scope of the present invention.

Claims

1. An automatic balancing method for an electromagnetic bearing-rotor system that does not require rotational speed, characterized in that, Includes the following steps: S1, Establish the mathematical model of the linear extended state observer, specifically as follows: The single control channel of the electromagnetic bearing-rotor system is simplified into a second-order system, described as follows: Where y is displacement, ξ is disturbance including unbalanced force of the same frequency, bu is control force generated by controller, λ1, λ2 and b represent the characteristics of second-order object, and b is composed of known part b0 and unknown part; When designing the controller, the total disturbance is defined as: Treating the total disturbance as the third state variable, the state equation of the linear extended state observer constructed for the controlled second-order object and the disturbance is: Wherein, the state vector z = [z1 z2 z3] T z1, z2 and z3 are the observed values ​​of displacement, displacement derivative and total disturbance of electromagnetic bearing-rotor system, respectively. A is the system matrix of the observer, B is the input matrix, C is the output matrix and L is the correction matrix. When the equation of state does not contain a physical model Where β1, β2, and β3 are undetermined coefficients obtained by configuring the observer poles; When the equation of state includes a physical model Where a0 is the observer stiffness and a1 is the observer damping; S2, Design the control structure of the linear extended state observer notch filter to attenuate the vibration caused by the disturbance, specifically: The input signal of the linear extended state observer is set as displacement y, and the output signal of the linear extended state observer is set as state variable z1. First, in the first discrete cycle, the input signal of the linear extended state observer is set as displacement y(1). Then, after one discrete cycle, the output signal is z1(1). In the second discrete cycle, the current displacement y(2) is subtracted from the output z1(1), and the result is used as the new input of the linear extended state observer. After one discrete cycle, the output signal is z1(2), which is used to subtract z1(2) from y(3) in the third discrete cycle. This process is repeated, and the signal after each subtraction is fed into the controller to maintain the stability of the system. S3, perform pole placement for the linearly extended state observer. The specific pole placement method is as follows: The poles of the linear extended state observer are divided into a pair of conjugate poles s1 and s2 that determine the resonant frequency and damping ratio, and a pole s3 that determines the tracking speed; the conjugate poles s1 and s2 are in the following form: Where η is the damping ratio, ω n is the resonant frequency, and j is the imaginary unit; this pair of poles causes the linear extended state observer to produce a resonant region at the resonant frequency; When the input of the linear extended state observer is displacement y and the output is the first state variable z1 of the state equation, within the frequency range below the resonance region, the output of the linear extended state observer is equal to the input; within the resonance region, the output of the linear extended state observer first increases and then decreases; within the frequency range above the resonance region, the output of the linear extended state observer is less than the input; the resonance frequency ω... n The damping ratio η is used to adjust the center position of the resonant region, and the damping ratio η is used to adjust the resonant region damping of the linear extended state observer. The third pole s3 has the following form: s3=-eηω n Where e is the error amplification ratio, used to reduce the tracking error of the linear extended state observer; When the frequency of the input signal exceeds the resonant frequency of the linear extended state observer, the output signal of the linear extended state observer cannot accurately track the input signal, and its output tends to decrease. When the pole s3 is placed further away in the left half-plane, the resonant frequency is not changed, but the output amplitude is increased, and the tracking error is reduced. S4, Embedded Physical Model The physical model can be embedded in the state equation of the linear extended state observer. The observer stiffness a0 is used to adjust the resonant frequency. Increasing the value of the observer stiffness a0 increases the resonant frequency, which has the same effect as increasing the resonant frequency of the poles. The observer damping a1 is used to adjust the resonance effect of the linear extended state observer. Decreasing the observer damping a1 enhances the negative gain at the resonant point, which has the same effect as decreasing the damping ratio η of the pole placement. By embedding the physical model, the value of the L matrix of the observer's mathematical model is reduced, thereby reducing the noise sensitivity.

2. The automatic balancing method as described in claim 1, characterized in that, The transfer function from the displacement input y to the running output z1 of the linear extended state observer is: Where s is the Laplace operator; the transfer function of the notch filter is: Due to the structure of the notch filter, the notch filter has a negative gain of at least 6 dB in the range below the observer's resonant region.

3. The automatic balancing method as described in claim 1, characterized in that, The effects of the pole placement method on the frequency characteristics of the linear extended state observer notch filter are as follows: The conjugate poles that determine the resonant frequency and damping ratio create a resonant region for the observer near the resonant frequency. Within this region, the amplitude of the notch filter first decreases and then increases with frequency, forming a negative gain region with a minimum value and a positive gain region with a maximum value. The frequency of the minimum value in the negative gain region is equal to the resonant frequency. The smaller the damping ratio, the more the minimum and maximum values ​​in the resonant region increase, the wider the resonant region becomes, and the wider the frequency band of the negative gain region within the resonant region becomes. The poles that determine the tracking speed affect the maximum value of the positive gain region in the resonant region; when the error amplification ratio is in the range of 0 to 5, the larger the error amplification ratio, the smaller the maximum value of the positive gain region in the resonant region; the phase characteristic in the resonant region starts from 0 at the resonant frequency and increases, then decreases back to 0.

4. The automatic balancing method as described in claim 1, characterized in that, When the frequency of the input signal is less than the resonant region, the notch filter has a base gain of -6dB. The gain decreases in the frequency range before the resonant frequency. Within the resonant region of the linear extended state observer notch filter, the amplitude-frequency characteristic forms a notch region, which acts as a notch filter. After the frequency exceeds the notch region, the gain of the notch filter gradually increases to 0 and loses its function.

5. The automatic balancing method as described in claim 1, characterized in that, When the frequency of the input signal is below the resonant frequency, the phase characteristic is 0. However, after passing through the observer bandwidth, the phase gradually increases and eventually returns to 0. Therefore, the linear extended state observer notch filter only reduces the gain of the frequency band below the observer bandwidth and the concave region near the bandwidth, and does not affect the phase of the controller.

6. The automatic balancing method as described in claim 1, characterized in that, The resonant frequency ω n The system is set to its rated operating frequency. The system will have optimal current suppression during rated operation and will also have suppression before accelerating to the rated speed.

7. The automatic balancing method as described in claim 1, characterized in that, The linear extended state observer has a finite gain on the input signal, the notch filter of the linear extended state observer has a finite gain at the resonant frequency, and the notch depth is adjustable.

8. The automatic balancing method as described in claim 1, characterized in that, The output of the linear extended state observer is then processed using the Fal nonlinear function to divide the total gap into a strong control region and a weak control region for automatic balancing. The boundary between the two control regions is determined by 30% of the total gap. The expression for the Fal function is: Where p is the input to the Fal function; δ is the linear domain, which is set as the strong control region for automatic balancing; α is the nonlinearity factor, which can be reduced to enhance the nonlinearity. The stronger the nonlinearity, the smaller the output in the weak control region. After applying the Fal function, for vibration signals exceeding the strong control region, an automatic attenuation and balancing output method is adopted to avoid excessive vibration and rubbing against the protective bearing.

9. The automatic balancing method as described in claim 1, characterized in that, The method does not require real-time speed information, can effectively reduce the vibration of the electromagnetic bearing-rotor system, and significantly reduce control current and base vibration. It does not require shutdown operation near the critical speed range.

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